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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something .
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and also , where did the `` operation '' symbol come from ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 .
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the denominator is 3x-2 would i still put positive 2 at the same place where sal puts the 3 0 in the video or does the 3x affect this ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division .
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so does this mean that every time you use synthetic division , there will be a remainder ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 .
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the x^5 term was 2 , and the x^4 term was 0 ( which became 6 ) , so how did two become the x^4 and 6 the x^3 ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ?
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i mean why ca n't we just use the number beside the x and then subtract the numbers on the second row from the numbers on the first row instead of add the two rows numbers altogether ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third .
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sal says that the coefficient to the x^4 term is 0 , would n't it be actually 1 ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division .
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how would you use synthetic division to divide a complex polynomial with a complex binomial ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say .
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so the last term is always the remainder ( if we have one ) right ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this .
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why is the last term of the polynomial the only one that is divided by x and another number ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division .
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how would you find the zeros of a function using synthetic division ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 .
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if i were to divide 3x^2 - 4x + 7 by x - 1 , i get 3x -1 with a remainder of 6 with long division and 3x - 7 with a remainder of 14 , can someone explain what i 'm doing wrong ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 .
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is there a way to factorise a polynomial with a degree greater than 2 ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division .
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what would happen if there was a coefficient next to x-3 example 2x-3 at the denominator ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it .
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do i have to learn synthetic division if i know how to do long division ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer .
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is there any difference between the two besides speed and space saving ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 .
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why did you use the zero ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 .
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hi , i was wondering why you would take the opposite of the constant in the denominator , since does n't that mean we are dividing by zero ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division .
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how do you use synthetic division when using binomials ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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so then i have my x to the fourth term . so it is 2x to the fourth . and we are done .
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why does 2x^5 become 2x^4 at the end of the video ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 .
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how do you apply the remainder theorem ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ?
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how would i divide synthetically if i have complex numbers ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say .
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how you find the number you divide from the polynomial equation ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 .
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where does the zero come from ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 .
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how do u do remainder theorem in polynomial functions ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division .
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what does sign of remainder - and + signify , it was - ( negative ) on previous and + now , how ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 .
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where did the zero ( 0 ) came from ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 .
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i do n't know how to do this type , no one seems to cover irrational numbers : x^4+2*squareroot ( 2 ) *x^3+4x^2+2*squareroot ( 2 ) *x+1 i know the answer is ( x^2+squareroot ( 2 ) *x+1 ) ^2 but how do you find the factors or how do you divide with synthetic division to get the perfect square factored form ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here .
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what do i do if there is no x + 3 being divided and i have to find a number to use ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 .
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how would you graph the remainder in a calculator with the rest of the equation ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 .
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so how would you go about this problem if ( x-3 ) was ( x^3 - x^2 ) ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division .
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how about ... x - 3 becomes 3x - 3 ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 .
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why is the answer a 4th degree polynomial when the question is a 5th degree ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division .
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where does the name 'synthetic division ' come from ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 .
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in college i took a situation in which we divide f ( x ) / ( x-a ) ( x-b ) by using synthetic division .. is there a lesson about it here ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 .
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how would i synthetically divide if my divisor variable had an exponential of more than 1 or a co-efficient ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 .
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what happens if you are dividing by a polynomial with a degree higher than 1 or more than two terms ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now .
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so if you are dividing by x+3 , do you multiply the coefficients by -3 and still add two numbers or subtract ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here .
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what do you do when a power of x is missing , say the problem is 5x^3 divided by x-3 , i know that you have to add a 0x2 and a 0x before dividing but do we also need to add a plain old zero as well as one of the coefficents ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division .
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is synthetic division only used for polynomials ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division .
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what i 'm asking is , why did the exponent change ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 .
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why do n't we use the 5th power , 3rd power , 2 powered throughout the video ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division .
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if you had an expression like 3x+5 where the coefficient is n't 1 , could you divide the expression by 3 to get x+5/3 and then do synthetic division ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 .
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so when you have x+4 you have to use the opposite of the sign ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 .
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what would you do if you had a problem where it has ( x+2 ) and a -1 as your exponent ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division .
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hod did you get zero as a coefficient when finding the numbers to place for the synthetic division ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 .
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if the x has an exponent else than 1 then the remainder will have x to some power , right ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 .
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then how do we decide what power the remainder 's x should have ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here .
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in x+3 you need to substitute x to a number that would result to zero , what if the x has a number in it for example : 5x+3 , how do you find out the divisor if this is the case ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division .
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why does the -3 turn into a 3 for dividing ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 .
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why was the second number a 0 ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 .
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how about if the denominator is 3y-1 ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division .
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will an expression like 3x-3 divide another expression just as x-3 would ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it .
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how do we find the factors of a cubic polynomial without long and synthetic division ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division .
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does the polynomial have to be in descending order ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 .
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what if you 're using synthetic division to find a f ( x ) that has x as any number greater than 1 ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division .
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so when you are using synthetic division , are you supposed to subtract or add when doing the carrying down and multiplying over ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division .
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how do you get no solutions for synthetic division ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division .
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what happens when the denominator has polynomials ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third .
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how do you know whether you put an addition or subtraction sign into your solution ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 .
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is there any way that there would n't be a remainder unless the constant was zero ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division .
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why is the 3 on the outside positive suddenly ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division .
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how do you figure out what is below the equation if nothing is there ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 .
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if the remainder is not zero , should i just keep going with the same number i 've been using to synthetically divide ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 .
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how do you divide a polynomial if the divisor is something like x^2+2x-1 ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division .
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is there any situation where you would have to use synthetic division instead of standard division ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say .
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what is the easiest way to pull out a root thats a whole number ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something .
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just wondering , would this be considered an easier way to divide polynomials ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here .
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what is the best place to get pie ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here .
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how do you get the zero in the problem ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this .
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what happened to the x^5 term ?
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actually try to simplify this rational expression . so let 's think about this step by step . so the first thing i want to do is write all of the coefficients of the numerator . so i have a 2 . oh , i have to be careful here . because the 2 is the coefficient for x to the fifth , i have no x to the fourth term . let me start over . so i have the 2 from 2x to the fifth . and then i have no x to the fourth . so it 's really 0x to the fourth . so i 'll put a 0 as the coefficient for the x to the fourth term . and then i have a negative 1 times x to the third . and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . i just have a positive 7 . and now let me just draw my little funky synthetic division operator-looking symbol . and remember , the type of synthetic division we 're doing , it only applies when we are dividing by an x plus or minus something . there 's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared . this only works when we have x plus or minus something . in this case we have x minus 3 . so we have the negative 3 here . and the process we show -- there 's other ways of doing it -- is you take the negative of this . so the negative of negative 3 is positive 3 . and now we 're ready to perform our synthetic division . so we 'll bring down this 2 and then multiply the 2 times the 3 . 2 time 3 gives us 6 . 0 plus 6 is 6 . and then we multiply that times the 3 , and we get positive 18 . negative 1 plus 18 is 17 . multiply that times the 3 . 17 times 3 is 51 . 3 plus 51 is 54 . multiply that times 3 . the numbers are getting kind of large now . so that 's going to be what ? 50 times 3 is 150 . 4 times 3 is 12 . so this is going to be 162 . negative 2 plus 162 is 160 . and then finally , 160 times 3 is going to be 480 . and you add 480 to 7 , and you get 487 . and you can think of it , i only have one term or one number to the left-hand side of this bar here . or i 'm just doing the standard , traditional x plus or minus something version of synthetic division , i should say . so i can separate this out , and now i 've essentially gotten my answer . and it looks like voodoo , and it kind of is voodoo . and that 's why i do n't like to do it , because you 're just memorizing an algorithm . but there are other videos why we explain why . and it can be fast and convenient and paper saving very often , like you see right here . but then we have our final answer . it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . and so this is our constant term . and so you 're going to have plus 160 plus 487 over x minus 3 . now this is our x term . so it 's going to be 54x plus all of this . this is going to be our x squared term . so this is going to be 17x squared plus 54x plus 160 and all of that . then this is going to be x to the third term . so this is going to be 6x to the third plus all of that . and then finally , this is our x to the fourth term -- 2x to the fourth . and let me erase this . so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . this thing simplifies to this right over here . and i encourage you to verify it with traditional algebraic long division .
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let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it .
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is it possible to use synthetic division in a long division problem ?
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hi , i ’ m john green and this is the final episode of crash course : world history , not because we ’ ve reached the end of history but because we ’ ve reached the particular middle where i happen to be living . today we ’ ll be considering whether globalization is a good thing , and along the way we ’ ll try to do something that you may not be used to doing in history classes : imagining the future . past john : mr. green , mr. green ! in the future , i ’ m gon na get to second base with molly brown . present john : no you won ’ t , me from the past , but the fact that when asked to imagine the future , you imagine your future says a lot about the contemporary world . and listen , me from the past , while there ’ s no question that your solipsistic individualism is bad both for you and for our species , the broader implications of individualism in general are a lot more complex . [ theme music ] man , i ’ m gon na miss you , intro . so last week ( ta-da ) we discussed how global economic interdependence has led , on average , to longer , healthier , more prosperous lives for humans -- not to mention an astonishing change in the overall human population . in the west , globalization has also led to the rise of a service economy . in the us and europe , most people now work not in agriculture or manufacturing but in some kind of service sector : healthcare , retail , education , entertainment , information technology , internet videos about world history , etc . and that switch has really changed our psychology , especially the psychology of upper classes living in the industrialized world . i mean , to quote fredric jameson , “ we are ... so far removed from the realities of production and work that we inhabit a dream world of artificial stimuli and televised experience. ” think of it this way : if you had to kill a chicken every time you visited kfc , you would probably eat fewer chickens . another change of psychology : many historians-of-the-now note that globalization has also led to a celebration of individualism , particularly in the wake of the failures of the marxist collectivist utopias . the generation that lived through the depression and world war ii saw large-scale collectivist responses to both those crises . and they were responses that limited freedom . like , the military draft , for instance , which limited your freedom , you know , not to be a soldier . or the collectivization of health insurance seen in most of the post-war west , which limited your freedom to go bankrupt from health care costs . or also government programs like social security , which limit your freedom not to pay for old people ’ s retirement . but since the 1960s , the ascendant idea of personal freedom minimally limited by government intervention has become very powerful . even the catholic church was part of this new search for individual freedom , as the second vatican council relaxed church rules in ways that weakened central authority , made concessions to individual styles of worship , even said that people of different religions could go to heaven . what good is heaven if it ’ s gon na be full of protestants ? it ’ s just gon na be like minnesota . so here in the last episode of crash course world history , in the last thirty seconds , i have offended , uh , 5/6ths of the world ’ s population in the form of non-catholics and , uh , all republicans , and probably some political moderates . who are confused about what obama ’ s healthcare law will and will not do . stan , maybe i should just make this episode just an extended rant where i reveal all of my political biases . and also my personal biases . look , you ’ re never gon na meet a historian who doesn ’ t have biases . but good historians try to acknowledge their biases and i am biased toward canada and its awesome healthcare system . i can ’ t lie . i ’ m very jealous of you guys . but perhaps the greatest effect of the victory of individualism was on sex and the family . we haven ’ t talked much about sex because my brother ’ s teaching biology , which is basically just sex , but sex is pretty important historically because it ’ s how we keep happening . but , in the 20th century , greater variety and availability of contraception made it possible for people to experiment with multiple sexual partners and helped to uncouple sex from child bearing , which was awesome , but individualism also had a destabilizing effect on families . as the great leo tolstoy put it , all happy families are alike , but each unhappy family is unhappy in its own way . but when your individual fulfillment trumps all , you needn ’ t live amid your uniquely unhappy family , you can just leave ! so , divorce rates have skyrocketed in the past few decades , and not just in the us . by the turn of the 21st century , divorce rates in china reached nearly 25 % , with 70 % of those divorces initiated by women . technology has also driven families apart , as parents and children spend increasing time alone in front of their individual screens , sharing fewer experiences . that ’ s individualism , too , but not of a kind that we usually celebrate . but probably the biggest consequence of globalization and the ensuing rise in human population has been humanity 's effect on the environment . while populations have increased partly thanks to better yields from existing farmland , much more land has also been brought under cultivation in the past half-century . often this meant cutting down trees in valuable rainforests– the best known example of this is what ’ s going on in the amazon , but it happens worldwide . and we 're losing land not just for food , but also to grow the global economy . oh , it ’ s time for the open letter ? an open letter to flowers . but first , let ’ s see what ’ s in the secret compartment today . oh , it ’ s fake flowers . thank you , stan . one for behind each ear . dear flowers , you capture the best and the worst of the globalized economy . you ’ re so pretty . even the fake ones are pretty . but the real one are constantly dying . they ’ ve got to be harvested , and shipped , and cut very efficiently . and it ’ s a global phenomenon . like there are flowers in my corner market from africa . these are from china , but because they are plastic , they could just be shipped in a shipping container . more people can afford to apologize by giving their romantic partners professionally cut and arranged roses than in any time in human history , but in that we have lost something , which is that the whole idea of flowers is that you had to go out into the field and , like , cut them and arrange them yourself to apologize . it ’ s not supposed to be , “ i ’ m sorry i forgot your birthday . here ’ s $ 8 worth of work that was done in kenya. ” it ’ s supposed to be , “ i ’ m sorry i forgot your birthday , so i went into the frakking forest and got you some frakking flowers . '' anyway , flowers , best wishes , john green aww ... you guys got me flowers for my last episode of world history . okay , let ’ s go to the thought bubble . as worldwide production and consumption increases , we use more resources , especially water and fossil fuels . globalization has made the average human richer , and rich people tend to use more of everything but especially energy . this has already resulted in climate change , which will likely accelerate . the global economy isn ’ t a zero-sum game . like , i don ’ t need to become more poor in order for someone else to become more rich . but growth , at least so far , has been dependent upon unsustainable use of the planet 's resources . the planet can ’ t sustain seven billion automobiles , for instance , or seven billion frequent flyers , although most of us who can afford to drive or fly feel entitled to do so . you 'll remember that when we talked about the industrial revolution , we discussed the virtuous cycle of more efficiency making things cheaper , which in turn made them easier to buy , which increased demand , which increased efficiency . but from the perspective of the planet , each turn in that cycle takes something : more land under cultivation , more carbon emissions , more resource extraction . that can ’ t go on forever , but worryingly , our current models of economic growth don ’ t allow for any other way . thanks , thought bubble . and then there is our astonishingly robust health . although much of the world has been ravaged by hiv/aids for the past three decades , there ’ s been a relative lack of global pandemics since the 1918 flu . and that ’ s particularly surprising given increased population density and more travel between population centers . china has seen 150 million people leave the countryside for cities in the last 20 years . this was shanghai in 1990 ; and this is shanghai in 2010 . the population of lagos was 41,000 in 1900 ; today , it 's almost 8 million . of course , people have been moving from country to city for a long time ; remember gilgamesh ? but the pace of that change has dramatically accelerated . similarly , there 's nothing new about international trade , but its pace has also increased dramatically : in 1960 , trade accounted for 24 % of the world 's gdp ; today , it ’ s more than double that . almost no human being alive today lives with stuff only manufactured in their home country , but a thousand years ago , only the richest of the rich could benefit from the silk road . still , trade isn ’ t new . and while it ’ s tempting to say that the types of goods being traded-– pharmaceuticals , computers , software , financial services -- represent something wholly new , you could just as easily see this as part of the evolution of trade itself . at some point silk was seen as a new trade good . as tastes change and consumers become more affluent , the things that they want to buy change . so is anything really different , or is it all just accelerated ? well , some historians argue that an economically interdependent world is much less likely to go to war . and that may be true , but increasing global , cultural , and economic integration hasn ’ t led to an end to violence . i mean , we 've seen large scale ethnic and nationalistic violence from rwanda to the former yugoslavia to the democratic republic of congo to afghanistan . globalization has not rid the world of violence . but there is an ideological shift in the age of globalization that does seem pretty new , and that ’ s the turn to democracy . now this isn ’ t the limited democracy of the ancient greeks , or the quirky republican system originally developed in the u.s. ; there are almost as many kinds of democracies as there are nations experiencing democracy . the fact is , however , that democracy and political freedom , especially the freedom to participate in and influence the government , have been on the rise all over the world since the 1980s and especially since 1990 . for instance , if you looked at the governments of most latin american countries during most of the 20th centuries , you would usually find them ruled by military strongmen . now , with a couple of exceptions ( fidel , hugo ) … stan , are they behind me right now ? because if they ’ re behind me , i am in favor of collectivizing oil revenue and distributing it to the poor . if they ’ re not behind me , that ’ s a terrible idea . right , but anyway , democracy is now flourishing in most of latin america . probably the most famous democratic success story is south africa , which jettisoned decades of apartheid in the 1990s and elected former dissident nelson mandela as its first black president in 1994 . it also adopted one of the most progressive constitutions in the world . but it ’ s worth remembering that democracy and economic success don ’ t always go hand in hand , as much as some americans wish they would . many new african democracies continue to struggle , the same is true in some latin american countries , and china has shown that you don ’ t need democracy in order to experience economic growth . but for a few countries , especially brazil and india , the combination of democracy and economic liberalism has unleashed impressive growth that has lifted millions out of poverty . so can we say that it 's good , then ? can we celebrate globalization , in spite of its destabilizing effects on families and the environment ? well , here 's where we have to imagine the future , because if some superbug shows up tomorrow and it travels through all these global trade routes and kills every living human , then globalization will have been very bad for human history : specifically , by ending it . if climate change continues to accelerate and displaces billions of people and causes widespread famines and flooding , then we will remember this period of human history as short-sighted , self-indulgent , and tremendously destructive . on the other hand , if we discover an asteroid hurdling toward earth and mobilize global industry and technology in such a way that we lose bruce willis but save the world , then globalization will be celebrated for millennia . i mean , assuming we have millennia and can convince bruce willis to go . in short , to understand the present , we have to imagine the future . that 's the thing about history , it depends on where you 're standing . from where i 'm standing , globalization has been a net positive , but then again , it 's been a pretty good run for heterosexual males of european descent . critics of globalization point out that billions have n't benefited much if at all from all this economic prosperity , and that the polarization of wealth is growing both within and across nations . and those criticisms are valid and they are troubling , but they aren ’ t new . disparities between those who have more and those who have less have existed pretty much from the moment agriculture enabled us to accumulate a surplus . at some times this inequality has been a big concern , as it was with jesus and muhammad , at other times not so much . inequalities are as old as human history , and almost as old is the debate about them . one thing that is new , however , is our ability to learn about them , to discuss them , and hopefully to find solutions for them together as a global community that is better integrated and more connected than it has ever been before . because here 's the other thing about history : you are making it . that old idea that history is the deeds of great men ? that was wrong . celebrated individuals do shape history , but so do the rest of us . and while it 's true that many historical forces -- malaria , meteors from space -- are n't human , it 's also true that every human is a historical force . you are changing the world every day . and it is our hope that by looking at the history that was made before us , we can see our own crucial decisions in a broader context . and i believe that context can help us make better choices , and better changes . thanks for watching . but , there ’ s no need to despair , crash course fans , i ’ ll see you next week for the beginning of our mini series on literature . crash course is produced and directed by stan muller . our script supervisor is meredith danko . the associate producer is danica johnson . the show is written by my high school history teacher , raoul meyer , and myself . and our graphics team is thought bubble . last week ’ s phrase of the week was `` cookie monster '' . this week ’ s phrase of the week was `` bruce willis , '' which i am telling you because we are retiring the idea of the phrase of the week . thank you so much for watching crash course : world history . it has been super fun to try to tell the history of the world in 42 twelve-minute videos . i hope you enjoyed it and i hope you ’ ll hang around for literature . thanks for watching , and as we say in my hometown , do n't forget to be awesome .
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anyway , flowers , best wishes , john green aww ... you guys got me flowers for my last episode of world history . okay , let ’ s go to the thought bubble . as worldwide production and consumption increases , we use more resources , especially water and fossil fuels . globalization has made the average human richer , and rich people tend to use more of everything but especially energy .
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during the thought bubble , he said that the use of lots of fossil fuels leads to climate change , but does n't the use of fossil fuels lead to global warming , not climate change ?
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hi , i ’ m john green and this is the final episode of crash course : world history , not because we ’ ve reached the end of history but because we ’ ve reached the particular middle where i happen to be living . today we ’ ll be considering whether globalization is a good thing , and along the way we ’ ll try to do something that you may not be used to doing in history classes : imagining the future . past john : mr. green , mr. green ! in the future , i ’ m gon na get to second base with molly brown . present john : no you won ’ t , me from the past , but the fact that when asked to imagine the future , you imagine your future says a lot about the contemporary world . and listen , me from the past , while there ’ s no question that your solipsistic individualism is bad both for you and for our species , the broader implications of individualism in general are a lot more complex . [ theme music ] man , i ’ m gon na miss you , intro . so last week ( ta-da ) we discussed how global economic interdependence has led , on average , to longer , healthier , more prosperous lives for humans -- not to mention an astonishing change in the overall human population . in the west , globalization has also led to the rise of a service economy . in the us and europe , most people now work not in agriculture or manufacturing but in some kind of service sector : healthcare , retail , education , entertainment , information technology , internet videos about world history , etc . and that switch has really changed our psychology , especially the psychology of upper classes living in the industrialized world . i mean , to quote fredric jameson , “ we are ... so far removed from the realities of production and work that we inhabit a dream world of artificial stimuli and televised experience. ” think of it this way : if you had to kill a chicken every time you visited kfc , you would probably eat fewer chickens . another change of psychology : many historians-of-the-now note that globalization has also led to a celebration of individualism , particularly in the wake of the failures of the marxist collectivist utopias . the generation that lived through the depression and world war ii saw large-scale collectivist responses to both those crises . and they were responses that limited freedom . like , the military draft , for instance , which limited your freedom , you know , not to be a soldier . or the collectivization of health insurance seen in most of the post-war west , which limited your freedom to go bankrupt from health care costs . or also government programs like social security , which limit your freedom not to pay for old people ’ s retirement . but since the 1960s , the ascendant idea of personal freedom minimally limited by government intervention has become very powerful . even the catholic church was part of this new search for individual freedom , as the second vatican council relaxed church rules in ways that weakened central authority , made concessions to individual styles of worship , even said that people of different religions could go to heaven . what good is heaven if it ’ s gon na be full of protestants ? it ’ s just gon na be like minnesota . so here in the last episode of crash course world history , in the last thirty seconds , i have offended , uh , 5/6ths of the world ’ s population in the form of non-catholics and , uh , all republicans , and probably some political moderates . who are confused about what obama ’ s healthcare law will and will not do . stan , maybe i should just make this episode just an extended rant where i reveal all of my political biases . and also my personal biases . look , you ’ re never gon na meet a historian who doesn ’ t have biases . but good historians try to acknowledge their biases and i am biased toward canada and its awesome healthcare system . i can ’ t lie . i ’ m very jealous of you guys . but perhaps the greatest effect of the victory of individualism was on sex and the family . we haven ’ t talked much about sex because my brother ’ s teaching biology , which is basically just sex , but sex is pretty important historically because it ’ s how we keep happening . but , in the 20th century , greater variety and availability of contraception made it possible for people to experiment with multiple sexual partners and helped to uncouple sex from child bearing , which was awesome , but individualism also had a destabilizing effect on families . as the great leo tolstoy put it , all happy families are alike , but each unhappy family is unhappy in its own way . but when your individual fulfillment trumps all , you needn ’ t live amid your uniquely unhappy family , you can just leave ! so , divorce rates have skyrocketed in the past few decades , and not just in the us . by the turn of the 21st century , divorce rates in china reached nearly 25 % , with 70 % of those divorces initiated by women . technology has also driven families apart , as parents and children spend increasing time alone in front of their individual screens , sharing fewer experiences . that ’ s individualism , too , but not of a kind that we usually celebrate . but probably the biggest consequence of globalization and the ensuing rise in human population has been humanity 's effect on the environment . while populations have increased partly thanks to better yields from existing farmland , much more land has also been brought under cultivation in the past half-century . often this meant cutting down trees in valuable rainforests– the best known example of this is what ’ s going on in the amazon , but it happens worldwide . and we 're losing land not just for food , but also to grow the global economy . oh , it ’ s time for the open letter ? an open letter to flowers . but first , let ’ s see what ’ s in the secret compartment today . oh , it ’ s fake flowers . thank you , stan . one for behind each ear . dear flowers , you capture the best and the worst of the globalized economy . you ’ re so pretty . even the fake ones are pretty . but the real one are constantly dying . they ’ ve got to be harvested , and shipped , and cut very efficiently . and it ’ s a global phenomenon . like there are flowers in my corner market from africa . these are from china , but because they are plastic , they could just be shipped in a shipping container . more people can afford to apologize by giving their romantic partners professionally cut and arranged roses than in any time in human history , but in that we have lost something , which is that the whole idea of flowers is that you had to go out into the field and , like , cut them and arrange them yourself to apologize . it ’ s not supposed to be , “ i ’ m sorry i forgot your birthday . here ’ s $ 8 worth of work that was done in kenya. ” it ’ s supposed to be , “ i ’ m sorry i forgot your birthday , so i went into the frakking forest and got you some frakking flowers . '' anyway , flowers , best wishes , john green aww ... you guys got me flowers for my last episode of world history . okay , let ’ s go to the thought bubble . as worldwide production and consumption increases , we use more resources , especially water and fossil fuels . globalization has made the average human richer , and rich people tend to use more of everything but especially energy . this has already resulted in climate change , which will likely accelerate . the global economy isn ’ t a zero-sum game . like , i don ’ t need to become more poor in order for someone else to become more rich . but growth , at least so far , has been dependent upon unsustainable use of the planet 's resources . the planet can ’ t sustain seven billion automobiles , for instance , or seven billion frequent flyers , although most of us who can afford to drive or fly feel entitled to do so . you 'll remember that when we talked about the industrial revolution , we discussed the virtuous cycle of more efficiency making things cheaper , which in turn made them easier to buy , which increased demand , which increased efficiency . but from the perspective of the planet , each turn in that cycle takes something : more land under cultivation , more carbon emissions , more resource extraction . that can ’ t go on forever , but worryingly , our current models of economic growth don ’ t allow for any other way . thanks , thought bubble . and then there is our astonishingly robust health . although much of the world has been ravaged by hiv/aids for the past three decades , there ’ s been a relative lack of global pandemics since the 1918 flu . and that ’ s particularly surprising given increased population density and more travel between population centers . china has seen 150 million people leave the countryside for cities in the last 20 years . this was shanghai in 1990 ; and this is shanghai in 2010 . the population of lagos was 41,000 in 1900 ; today , it 's almost 8 million . of course , people have been moving from country to city for a long time ; remember gilgamesh ? but the pace of that change has dramatically accelerated . similarly , there 's nothing new about international trade , but its pace has also increased dramatically : in 1960 , trade accounted for 24 % of the world 's gdp ; today , it ’ s more than double that . almost no human being alive today lives with stuff only manufactured in their home country , but a thousand years ago , only the richest of the rich could benefit from the silk road . still , trade isn ’ t new . and while it ’ s tempting to say that the types of goods being traded-– pharmaceuticals , computers , software , financial services -- represent something wholly new , you could just as easily see this as part of the evolution of trade itself . at some point silk was seen as a new trade good . as tastes change and consumers become more affluent , the things that they want to buy change . so is anything really different , or is it all just accelerated ? well , some historians argue that an economically interdependent world is much less likely to go to war . and that may be true , but increasing global , cultural , and economic integration hasn ’ t led to an end to violence . i mean , we 've seen large scale ethnic and nationalistic violence from rwanda to the former yugoslavia to the democratic republic of congo to afghanistan . globalization has not rid the world of violence . but there is an ideological shift in the age of globalization that does seem pretty new , and that ’ s the turn to democracy . now this isn ’ t the limited democracy of the ancient greeks , or the quirky republican system originally developed in the u.s. ; there are almost as many kinds of democracies as there are nations experiencing democracy . the fact is , however , that democracy and political freedom , especially the freedom to participate in and influence the government , have been on the rise all over the world since the 1980s and especially since 1990 . for instance , if you looked at the governments of most latin american countries during most of the 20th centuries , you would usually find them ruled by military strongmen . now , with a couple of exceptions ( fidel , hugo ) … stan , are they behind me right now ? because if they ’ re behind me , i am in favor of collectivizing oil revenue and distributing it to the poor . if they ’ re not behind me , that ’ s a terrible idea . right , but anyway , democracy is now flourishing in most of latin america . probably the most famous democratic success story is south africa , which jettisoned decades of apartheid in the 1990s and elected former dissident nelson mandela as its first black president in 1994 . it also adopted one of the most progressive constitutions in the world . but it ’ s worth remembering that democracy and economic success don ’ t always go hand in hand , as much as some americans wish they would . many new african democracies continue to struggle , the same is true in some latin american countries , and china has shown that you don ’ t need democracy in order to experience economic growth . but for a few countries , especially brazil and india , the combination of democracy and economic liberalism has unleashed impressive growth that has lifted millions out of poverty . so can we say that it 's good , then ? can we celebrate globalization , in spite of its destabilizing effects on families and the environment ? well , here 's where we have to imagine the future , because if some superbug shows up tomorrow and it travels through all these global trade routes and kills every living human , then globalization will have been very bad for human history : specifically , by ending it . if climate change continues to accelerate and displaces billions of people and causes widespread famines and flooding , then we will remember this period of human history as short-sighted , self-indulgent , and tremendously destructive . on the other hand , if we discover an asteroid hurdling toward earth and mobilize global industry and technology in such a way that we lose bruce willis but save the world , then globalization will be celebrated for millennia . i mean , assuming we have millennia and can convince bruce willis to go . in short , to understand the present , we have to imagine the future . that 's the thing about history , it depends on where you 're standing . from where i 'm standing , globalization has been a net positive , but then again , it 's been a pretty good run for heterosexual males of european descent . critics of globalization point out that billions have n't benefited much if at all from all this economic prosperity , and that the polarization of wealth is growing both within and across nations . and those criticisms are valid and they are troubling , but they aren ’ t new . disparities between those who have more and those who have less have existed pretty much from the moment agriculture enabled us to accumulate a surplus . at some times this inequality has been a big concern , as it was with jesus and muhammad , at other times not so much . inequalities are as old as human history , and almost as old is the debate about them . one thing that is new , however , is our ability to learn about them , to discuss them , and hopefully to find solutions for them together as a global community that is better integrated and more connected than it has ever been before . because here 's the other thing about history : you are making it . that old idea that history is the deeds of great men ? that was wrong . celebrated individuals do shape history , but so do the rest of us . and while it 's true that many historical forces -- malaria , meteors from space -- are n't human , it 's also true that every human is a historical force . you are changing the world every day . and it is our hope that by looking at the history that was made before us , we can see our own crucial decisions in a broader context . and i believe that context can help us make better choices , and better changes . thanks for watching . but , there ’ s no need to despair , crash course fans , i ’ ll see you next week for the beginning of our mini series on literature . crash course is produced and directed by stan muller . our script supervisor is meredith danko . the associate producer is danica johnson . the show is written by my high school history teacher , raoul meyer , and myself . and our graphics team is thought bubble . last week ’ s phrase of the week was `` cookie monster '' . this week ’ s phrase of the week was `` bruce willis , '' which i am telling you because we are retiring the idea of the phrase of the week . thank you so much for watching crash course : world history . it has been super fun to try to tell the history of the world in 42 twelve-minute videos . i hope you enjoyed it and i hope you ’ ll hang around for literature . thanks for watching , and as we say in my hometown , do n't forget to be awesome .
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and while it 's true that many historical forces -- malaria , meteors from space -- are n't human , it 's also true that every human is a historical force . you are changing the world every day . and it is our hope that by looking at the history that was made before us , we can see our own crucial decisions in a broader context .
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is there some way we can reduce/stop global warming , at home , every day , by doing small things ?
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i think we 're now ready to tackle the big picture and what has our government officials so worried right now . so what i 've done is , i 've just drawn the balance sheets for a bunch of banks . obviously , this is simplified . and i made all of their balance sheets look the same . all of these banks , each of these kind of represents the balance sheet of a bank . and just to explain it , the left-hand side of this balance sheet , so this column right here -- and maybe i can , at least for the first bank , mark it a little bit . so what i 'm squaring off in magenta , that 's the assets of that bank . what i 'm squaring off in blue , that 's the liabilities of the bank . and what i wrote here is , it has $ 4 billion of liabilities . its assets , i divided it between $ 3 billion of other assets and $ 2 billion of cdos . because we want to focus on the cdos , because that 's the crux of everything that 's going on . and we have $ 5 billion in assets , $ 4 billion of liabilities , so you have $ 1 billion in equity . so that 's what 's left there . so this is just another visual representation that liabilities plus equity is equal to assets . or assets minus liabilities is equal to equity . and i 've just copied and pasted this one balance sheet a bunch of times . i do n't know whether we 're going to use all those . but let 's just assume , for simplicity , that a ton of banks in the system have this identical balance sheet . obviously , they do n't have an identical balance sheet . but all of their balance sheets might have kind of similar properties . this is n't always the case , different banks have different exposures to cdos . some of them have a lot , some of them have a little bit . some of them are valuing them more conservatively than others . but just for the sake of simplicity , i 've just made all the banks in the situation where the book value of the cdos that they have on their balance sheets is the larger than their equity value . and i did that for a reason . because it leads to the issue of , are these banks facing just a liquidity issue or are they facing just a solvency issue ? if you believe that these are worth $ 3 billion , these assets , these liabilities are worth $ 4 billion , then the crux of whether it 's a liquidity or a solvency issue all falls down as to whether these are worth $ 2 billion or not . for example , if these are worth $ 2 billion , then you have $ 1 billion of equity . if these are worth $ 1.5 billion , well maybe they 're being a little optimistic here , but you 'll still have $ 0.5 billion of equity . so you 're still solvent . and in that situation , in theory , one is just if they do n't have the cash when some of their debt comes due , they should just be able to borrow some money and get past that hurdle . and then in the future maybe sell their assets and still have positive equity . however , if the true value of those cdos , and this is kind of a philosophical question , what 's the true value of anything ? and the best thing that we as humans have been to be able to come up with is a market . the market value tends to be the best representation of the true value of something . let 's say the true value of this is $ 1 billion or less , then we have a situation . for example , if these are worth nothing , then we only have $ 3 billion of assets , $ 4 billion of liabilities , we have negative equity . this company is worth nothing . and to lend this bank or this company any money would just be throwing good money after bad . because that money is just going to go into a black hole . because one of the people who this company owes money to is probably not going to see their money . and if you are the most junior person lending the money -- which means that when all the money is distributed if they go into bankruptcy , you 're the last person to see the money -- then you 're just throwing good money after bad . so that 's the issue . but i want you to see the big picture now . because if it was just an issue with one bank it would n't be a big deal . if it was just bear stearns or if it was just lehman brothers , not a big deal , let the greedy bankers go bankrupt . and they probably are doing just fine with the bonuses they 've collected after sourcing these cdos for the past eight years or five years or however long . but what i want to show you in this video is what people are talking about when they say systemic risk . so these $ 4 billion in liabilities , these are loans , maybe from other banks . in fact , probably from other banks . and those loans from other banks , those are assets of other banks . for example , let 's say this is bank a , this is bank b . maybe a billion of these are a loan from bank b . and if this is a loan from bank b , bank b would have an asset called loan to bank a . on bank b 's balance sheet we 're calling this a loan to bank a . this is one of its assets . and then one of its liabilities will be a loan from bank b . so how can i say this ? they took this money and they gave it to b. i 'm sorry , b had money , gave it to a in the form of a loan . and so that cash ended up here . and they got an asset called loan to bank a . and this is a liability , loan from bank b . and they might have taken that money and they might have lent it to bank c down here . i think you 're starting to see how this gets pretty hairy very fast . so let 's say that bank a , one of its $ 3 billion in assets , is a loan to bank c. and so on bank c 's balance sheet , it 'll say loan from bank a . or so we owe a $ 1 billion . and a says , c owes me $ 1 billion , and that 's all fine . and then you see that oh , we owe b $ 1 billion . and then we could keep doing this . or i could just even make this into a circle already . so maybe bank b has some money that it owes to someone else . and let 's say that someone else , just for fun , just to make this interesting -- i think you can extrapolate and think about how this gets complicated very fast . bank b has borrowed money from bank c. so bank c will have an asset here that says , no i lent money to bank b . fair enough . ok , so now we 're in an interesting situation . let 's say this loan , the loan from bank b to bank a comes due . and we 've studied this multiple times . and let 's say for whatever reason , all of these other loans , they 're not liquid . they 're not due yet . so bank a ca n't get rid of these loans . so let 's say this comes due , this is $ 4 billion . they ca n't sell any of this . so bank a has to come up with $ 1 billion somehow for bank b . so that 's the situation we 're dealing with . i 'm just going to say that they ca n't sell any of these assets . so it all comes down to the cdos . so there 's a couple of issues here . if you think it is just an issue of illiquidity , if these are $ 2 billion of assets , they 're really worth $ 2 billion , but bank a just ca n't sell them . because either there 's quote-unquote nobody willing to buy . although , i would argue if no-one is willing to buy something , then its true value is probably zero . but let 's just say bank a says no-one is willing to buy , we 're just illiquid , this is really worth $ 2 billion . so one situation is they could get a loan from someone . maybe the fed would be willing to take this as collateral . so they would give this as collateral to the fed . maybe the fed will give them a billion dollar loan . and then they can use that to pay bank b . let 's say that 's off the table because this is just smelly enough collateral that not even the fed , which we now realize is willing to do anything to support the markets , not even the fed is willing to give them a loan . or enough of a loan to pay off that loan . the other situation is maybe they can get an equity infusion from a sovereign wealth fund . and we covered that a couple of videos ago . where the sovereign wealth fund will essentially inject some cash . it 'll dilute the shares and then you know maybe we had 500 million shares before . now we 'll have 2 billion shares . so the sovereign wealth fund will take over roughly 80 % of the company . and in exchange for 80 % of the company , would give maybe $ 2 billion and then you could use that to pay off this loan . but let 's say that that 's not on the table anymore either . because the sovereign wealth funds have gotten burned so much . so what happens ? well we learned what happens . if you ca n't get a loan , a new loan , to replace this loan , or if you ca n't get an equity infusion from kind of a greater fool , what happens ? you go into bankruptcy . and this is what happened to lehman brothers . lehman brothers went into bankruptcy . no sovereign wealth fund , no one else bought the company . and i should probably do another video on that scenario . and they could n't get a loan . so they went bankrupt . i should call this company l actually . but i 'll call it company a for now . because i do n't want to impugn anyone . i actually do n't think lehman was any worse or better than any of the other players here . so when they go into bankruptcy , something very interesting happens . now , bank b , they were already worried about these cdos . these cdos were already an issue . and they were probably thinking , boy when when loan c comes due , i 'm going to be in trouble . or when loan d , or f , or whatever , i 'm going to be in trouble because i 'm going to be in that situation that i 'm essentially forcing bank a into right now . but now i have a new problem . this loan to bank a is n't getting paid off . and who knows ? bank a is going to go into bankruptcy . maybe in bankruptcy we realize that these are worth nothing . and if those are worth nothing , then maybe i 'm very junior in seniority in terms of where my loan is and maybe i get nothing . or i get a few pennies on the dollar here . so maybe i thought this was $ 1 billion and i have to write this down to $ 0.5 billion . so now i have two problems . i have this and i have this . and once again , this is a non liquid loan . bank a is in bankruptcy . and if i wanted to somehow get the value of this i have to wait for all of bank a 's assets to go into liquidation . and then whatever assets i get i would have to sell it . so this is kind of a frozen asset . so once again , i 'm stuck holding this non liquid asset . so now i have this non liquid asset that 's probably not worth what i thought it was , which was a loan to a . then i also have these cdos . and now , god forbid , let 's say that i had another loan to bank d. and now let 's say bank d goes bankrupt . and then i have another loan that 's bad on top of these cdos . but the cdos were the crux of the issue . that 's what caused the situation . if bank a could have only sold this cdo for $ 2 billion , it would n't have caused this chain reaction . and lehman brothers really was the thing that catalyzed this whole chain of events . and then you can imagine now bank c is worried because now bank b has all of these illiquid assets on top of these cdos and it starts to look bad . and you can imagine , now it 's even less likely that when a bank , let 's say that bank d is the next one to go into a dire situation , it 's even less likely that bank d can get a loan from a third bank . because all the banks are getting scared now . all the banks are saying , i 'm not going to loan money to anyone . if i can get any cash from anybody i 'm just going to keep it . so that when it 's my turn , when the market starts looking at me , i at least have a little cash . so everyone is frozen . everyone wants to collect their loans from everyone else and no one wants to give loans to anybody else . so that 's the situation we 're in . and that 's the difficulty that the fed is somehow trying to unwind . and i realized i 'm out of time again . i will confront that issue in the next video .
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so these $ 4 billion in liabilities , these are loans , maybe from other banks . in fact , probably from other banks . and those loans from other banks , those are assets of other banks . for example , let 's say this is bank a , this is bank b .
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why do banks borrow from each other in a cycle like kahn suggested ?
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i think we 're now ready to tackle the big picture and what has our government officials so worried right now . so what i 've done is , i 've just drawn the balance sheets for a bunch of banks . obviously , this is simplified . and i made all of their balance sheets look the same . all of these banks , each of these kind of represents the balance sheet of a bank . and just to explain it , the left-hand side of this balance sheet , so this column right here -- and maybe i can , at least for the first bank , mark it a little bit . so what i 'm squaring off in magenta , that 's the assets of that bank . what i 'm squaring off in blue , that 's the liabilities of the bank . and what i wrote here is , it has $ 4 billion of liabilities . its assets , i divided it between $ 3 billion of other assets and $ 2 billion of cdos . because we want to focus on the cdos , because that 's the crux of everything that 's going on . and we have $ 5 billion in assets , $ 4 billion of liabilities , so you have $ 1 billion in equity . so that 's what 's left there . so this is just another visual representation that liabilities plus equity is equal to assets . or assets minus liabilities is equal to equity . and i 've just copied and pasted this one balance sheet a bunch of times . i do n't know whether we 're going to use all those . but let 's just assume , for simplicity , that a ton of banks in the system have this identical balance sheet . obviously , they do n't have an identical balance sheet . but all of their balance sheets might have kind of similar properties . this is n't always the case , different banks have different exposures to cdos . some of them have a lot , some of them have a little bit . some of them are valuing them more conservatively than others . but just for the sake of simplicity , i 've just made all the banks in the situation where the book value of the cdos that they have on their balance sheets is the larger than their equity value . and i did that for a reason . because it leads to the issue of , are these banks facing just a liquidity issue or are they facing just a solvency issue ? if you believe that these are worth $ 3 billion , these assets , these liabilities are worth $ 4 billion , then the crux of whether it 's a liquidity or a solvency issue all falls down as to whether these are worth $ 2 billion or not . for example , if these are worth $ 2 billion , then you have $ 1 billion of equity . if these are worth $ 1.5 billion , well maybe they 're being a little optimistic here , but you 'll still have $ 0.5 billion of equity . so you 're still solvent . and in that situation , in theory , one is just if they do n't have the cash when some of their debt comes due , they should just be able to borrow some money and get past that hurdle . and then in the future maybe sell their assets and still have positive equity . however , if the true value of those cdos , and this is kind of a philosophical question , what 's the true value of anything ? and the best thing that we as humans have been to be able to come up with is a market . the market value tends to be the best representation of the true value of something . let 's say the true value of this is $ 1 billion or less , then we have a situation . for example , if these are worth nothing , then we only have $ 3 billion of assets , $ 4 billion of liabilities , we have negative equity . this company is worth nothing . and to lend this bank or this company any money would just be throwing good money after bad . because that money is just going to go into a black hole . because one of the people who this company owes money to is probably not going to see their money . and if you are the most junior person lending the money -- which means that when all the money is distributed if they go into bankruptcy , you 're the last person to see the money -- then you 're just throwing good money after bad . so that 's the issue . but i want you to see the big picture now . because if it was just an issue with one bank it would n't be a big deal . if it was just bear stearns or if it was just lehman brothers , not a big deal , let the greedy bankers go bankrupt . and they probably are doing just fine with the bonuses they 've collected after sourcing these cdos for the past eight years or five years or however long . but what i want to show you in this video is what people are talking about when they say systemic risk . so these $ 4 billion in liabilities , these are loans , maybe from other banks . in fact , probably from other banks . and those loans from other banks , those are assets of other banks . for example , let 's say this is bank a , this is bank b . maybe a billion of these are a loan from bank b . and if this is a loan from bank b , bank b would have an asset called loan to bank a . on bank b 's balance sheet we 're calling this a loan to bank a . this is one of its assets . and then one of its liabilities will be a loan from bank b . so how can i say this ? they took this money and they gave it to b. i 'm sorry , b had money , gave it to a in the form of a loan . and so that cash ended up here . and they got an asset called loan to bank a . and this is a liability , loan from bank b . and they might have taken that money and they might have lent it to bank c down here . i think you 're starting to see how this gets pretty hairy very fast . so let 's say that bank a , one of its $ 3 billion in assets , is a loan to bank c. and so on bank c 's balance sheet , it 'll say loan from bank a . or so we owe a $ 1 billion . and a says , c owes me $ 1 billion , and that 's all fine . and then you see that oh , we owe b $ 1 billion . and then we could keep doing this . or i could just even make this into a circle already . so maybe bank b has some money that it owes to someone else . and let 's say that someone else , just for fun , just to make this interesting -- i think you can extrapolate and think about how this gets complicated very fast . bank b has borrowed money from bank c. so bank c will have an asset here that says , no i lent money to bank b . fair enough . ok , so now we 're in an interesting situation . let 's say this loan , the loan from bank b to bank a comes due . and we 've studied this multiple times . and let 's say for whatever reason , all of these other loans , they 're not liquid . they 're not due yet . so bank a ca n't get rid of these loans . so let 's say this comes due , this is $ 4 billion . they ca n't sell any of this . so bank a has to come up with $ 1 billion somehow for bank b . so that 's the situation we 're dealing with . i 'm just going to say that they ca n't sell any of these assets . so it all comes down to the cdos . so there 's a couple of issues here . if you think it is just an issue of illiquidity , if these are $ 2 billion of assets , they 're really worth $ 2 billion , but bank a just ca n't sell them . because either there 's quote-unquote nobody willing to buy . although , i would argue if no-one is willing to buy something , then its true value is probably zero . but let 's just say bank a says no-one is willing to buy , we 're just illiquid , this is really worth $ 2 billion . so one situation is they could get a loan from someone . maybe the fed would be willing to take this as collateral . so they would give this as collateral to the fed . maybe the fed will give them a billion dollar loan . and then they can use that to pay bank b . let 's say that 's off the table because this is just smelly enough collateral that not even the fed , which we now realize is willing to do anything to support the markets , not even the fed is willing to give them a loan . or enough of a loan to pay off that loan . the other situation is maybe they can get an equity infusion from a sovereign wealth fund . and we covered that a couple of videos ago . where the sovereign wealth fund will essentially inject some cash . it 'll dilute the shares and then you know maybe we had 500 million shares before . now we 'll have 2 billion shares . so the sovereign wealth fund will take over roughly 80 % of the company . and in exchange for 80 % of the company , would give maybe $ 2 billion and then you could use that to pay off this loan . but let 's say that that 's not on the table anymore either . because the sovereign wealth funds have gotten burned so much . so what happens ? well we learned what happens . if you ca n't get a loan , a new loan , to replace this loan , or if you ca n't get an equity infusion from kind of a greater fool , what happens ? you go into bankruptcy . and this is what happened to lehman brothers . lehman brothers went into bankruptcy . no sovereign wealth fund , no one else bought the company . and i should probably do another video on that scenario . and they could n't get a loan . so they went bankrupt . i should call this company l actually . but i 'll call it company a for now . because i do n't want to impugn anyone . i actually do n't think lehman was any worse or better than any of the other players here . so when they go into bankruptcy , something very interesting happens . now , bank b , they were already worried about these cdos . these cdos were already an issue . and they were probably thinking , boy when when loan c comes due , i 'm going to be in trouble . or when loan d , or f , or whatever , i 'm going to be in trouble because i 'm going to be in that situation that i 'm essentially forcing bank a into right now . but now i have a new problem . this loan to bank a is n't getting paid off . and who knows ? bank a is going to go into bankruptcy . maybe in bankruptcy we realize that these are worth nothing . and if those are worth nothing , then maybe i 'm very junior in seniority in terms of where my loan is and maybe i get nothing . or i get a few pennies on the dollar here . so maybe i thought this was $ 1 billion and i have to write this down to $ 0.5 billion . so now i have two problems . i have this and i have this . and once again , this is a non liquid loan . bank a is in bankruptcy . and if i wanted to somehow get the value of this i have to wait for all of bank a 's assets to go into liquidation . and then whatever assets i get i would have to sell it . so this is kind of a frozen asset . so once again , i 'm stuck holding this non liquid asset . so now i have this non liquid asset that 's probably not worth what i thought it was , which was a loan to a . then i also have these cdos . and now , god forbid , let 's say that i had another loan to bank d. and now let 's say bank d goes bankrupt . and then i have another loan that 's bad on top of these cdos . but the cdos were the crux of the issue . that 's what caused the situation . if bank a could have only sold this cdo for $ 2 billion , it would n't have caused this chain reaction . and lehman brothers really was the thing that catalyzed this whole chain of events . and then you can imagine now bank c is worried because now bank b has all of these illiquid assets on top of these cdos and it starts to look bad . and you can imagine , now it 's even less likely that when a bank , let 's say that bank d is the next one to go into a dire situation , it 's even less likely that bank d can get a loan from a third bank . because all the banks are getting scared now . all the banks are saying , i 'm not going to loan money to anyone . if i can get any cash from anybody i 'm just going to keep it . so that when it 's my turn , when the market starts looking at me , i at least have a little cash . so everyone is frozen . everyone wants to collect their loans from everyone else and no one wants to give loans to anybody else . so that 's the situation we 're in . and that 's the difficulty that the fed is somehow trying to unwind . and i realized i 'm out of time again . i will confront that issue in the next video .
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i think you 're starting to see how this gets pretty hairy very fast . so let 's say that bank a , one of its $ 3 billion in assets , is a loan to bank c. and so on bank c 's balance sheet , it 'll say loan from bank a . or so we owe a $ 1 billion .
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does bank c still need to pay off the loan ?
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i think we 're now ready to tackle the big picture and what has our government officials so worried right now . so what i 've done is , i 've just drawn the balance sheets for a bunch of banks . obviously , this is simplified . and i made all of their balance sheets look the same . all of these banks , each of these kind of represents the balance sheet of a bank . and just to explain it , the left-hand side of this balance sheet , so this column right here -- and maybe i can , at least for the first bank , mark it a little bit . so what i 'm squaring off in magenta , that 's the assets of that bank . what i 'm squaring off in blue , that 's the liabilities of the bank . and what i wrote here is , it has $ 4 billion of liabilities . its assets , i divided it between $ 3 billion of other assets and $ 2 billion of cdos . because we want to focus on the cdos , because that 's the crux of everything that 's going on . and we have $ 5 billion in assets , $ 4 billion of liabilities , so you have $ 1 billion in equity . so that 's what 's left there . so this is just another visual representation that liabilities plus equity is equal to assets . or assets minus liabilities is equal to equity . and i 've just copied and pasted this one balance sheet a bunch of times . i do n't know whether we 're going to use all those . but let 's just assume , for simplicity , that a ton of banks in the system have this identical balance sheet . obviously , they do n't have an identical balance sheet . but all of their balance sheets might have kind of similar properties . this is n't always the case , different banks have different exposures to cdos . some of them have a lot , some of them have a little bit . some of them are valuing them more conservatively than others . but just for the sake of simplicity , i 've just made all the banks in the situation where the book value of the cdos that they have on their balance sheets is the larger than their equity value . and i did that for a reason . because it leads to the issue of , are these banks facing just a liquidity issue or are they facing just a solvency issue ? if you believe that these are worth $ 3 billion , these assets , these liabilities are worth $ 4 billion , then the crux of whether it 's a liquidity or a solvency issue all falls down as to whether these are worth $ 2 billion or not . for example , if these are worth $ 2 billion , then you have $ 1 billion of equity . if these are worth $ 1.5 billion , well maybe they 're being a little optimistic here , but you 'll still have $ 0.5 billion of equity . so you 're still solvent . and in that situation , in theory , one is just if they do n't have the cash when some of their debt comes due , they should just be able to borrow some money and get past that hurdle . and then in the future maybe sell their assets and still have positive equity . however , if the true value of those cdos , and this is kind of a philosophical question , what 's the true value of anything ? and the best thing that we as humans have been to be able to come up with is a market . the market value tends to be the best representation of the true value of something . let 's say the true value of this is $ 1 billion or less , then we have a situation . for example , if these are worth nothing , then we only have $ 3 billion of assets , $ 4 billion of liabilities , we have negative equity . this company is worth nothing . and to lend this bank or this company any money would just be throwing good money after bad . because that money is just going to go into a black hole . because one of the people who this company owes money to is probably not going to see their money . and if you are the most junior person lending the money -- which means that when all the money is distributed if they go into bankruptcy , you 're the last person to see the money -- then you 're just throwing good money after bad . so that 's the issue . but i want you to see the big picture now . because if it was just an issue with one bank it would n't be a big deal . if it was just bear stearns or if it was just lehman brothers , not a big deal , let the greedy bankers go bankrupt . and they probably are doing just fine with the bonuses they 've collected after sourcing these cdos for the past eight years or five years or however long . but what i want to show you in this video is what people are talking about when they say systemic risk . so these $ 4 billion in liabilities , these are loans , maybe from other banks . in fact , probably from other banks . and those loans from other banks , those are assets of other banks . for example , let 's say this is bank a , this is bank b . maybe a billion of these are a loan from bank b . and if this is a loan from bank b , bank b would have an asset called loan to bank a . on bank b 's balance sheet we 're calling this a loan to bank a . this is one of its assets . and then one of its liabilities will be a loan from bank b . so how can i say this ? they took this money and they gave it to b. i 'm sorry , b had money , gave it to a in the form of a loan . and so that cash ended up here . and they got an asset called loan to bank a . and this is a liability , loan from bank b . and they might have taken that money and they might have lent it to bank c down here . i think you 're starting to see how this gets pretty hairy very fast . so let 's say that bank a , one of its $ 3 billion in assets , is a loan to bank c. and so on bank c 's balance sheet , it 'll say loan from bank a . or so we owe a $ 1 billion . and a says , c owes me $ 1 billion , and that 's all fine . and then you see that oh , we owe b $ 1 billion . and then we could keep doing this . or i could just even make this into a circle already . so maybe bank b has some money that it owes to someone else . and let 's say that someone else , just for fun , just to make this interesting -- i think you can extrapolate and think about how this gets complicated very fast . bank b has borrowed money from bank c. so bank c will have an asset here that says , no i lent money to bank b . fair enough . ok , so now we 're in an interesting situation . let 's say this loan , the loan from bank b to bank a comes due . and we 've studied this multiple times . and let 's say for whatever reason , all of these other loans , they 're not liquid . they 're not due yet . so bank a ca n't get rid of these loans . so let 's say this comes due , this is $ 4 billion . they ca n't sell any of this . so bank a has to come up with $ 1 billion somehow for bank b . so that 's the situation we 're dealing with . i 'm just going to say that they ca n't sell any of these assets . so it all comes down to the cdos . so there 's a couple of issues here . if you think it is just an issue of illiquidity , if these are $ 2 billion of assets , they 're really worth $ 2 billion , but bank a just ca n't sell them . because either there 's quote-unquote nobody willing to buy . although , i would argue if no-one is willing to buy something , then its true value is probably zero . but let 's just say bank a says no-one is willing to buy , we 're just illiquid , this is really worth $ 2 billion . so one situation is they could get a loan from someone . maybe the fed would be willing to take this as collateral . so they would give this as collateral to the fed . maybe the fed will give them a billion dollar loan . and then they can use that to pay bank b . let 's say that 's off the table because this is just smelly enough collateral that not even the fed , which we now realize is willing to do anything to support the markets , not even the fed is willing to give them a loan . or enough of a loan to pay off that loan . the other situation is maybe they can get an equity infusion from a sovereign wealth fund . and we covered that a couple of videos ago . where the sovereign wealth fund will essentially inject some cash . it 'll dilute the shares and then you know maybe we had 500 million shares before . now we 'll have 2 billion shares . so the sovereign wealth fund will take over roughly 80 % of the company . and in exchange for 80 % of the company , would give maybe $ 2 billion and then you could use that to pay off this loan . but let 's say that that 's not on the table anymore either . because the sovereign wealth funds have gotten burned so much . so what happens ? well we learned what happens . if you ca n't get a loan , a new loan , to replace this loan , or if you ca n't get an equity infusion from kind of a greater fool , what happens ? you go into bankruptcy . and this is what happened to lehman brothers . lehman brothers went into bankruptcy . no sovereign wealth fund , no one else bought the company . and i should probably do another video on that scenario . and they could n't get a loan . so they went bankrupt . i should call this company l actually . but i 'll call it company a for now . because i do n't want to impugn anyone . i actually do n't think lehman was any worse or better than any of the other players here . so when they go into bankruptcy , something very interesting happens . now , bank b , they were already worried about these cdos . these cdos were already an issue . and they were probably thinking , boy when when loan c comes due , i 'm going to be in trouble . or when loan d , or f , or whatever , i 'm going to be in trouble because i 'm going to be in that situation that i 'm essentially forcing bank a into right now . but now i have a new problem . this loan to bank a is n't getting paid off . and who knows ? bank a is going to go into bankruptcy . maybe in bankruptcy we realize that these are worth nothing . and if those are worth nothing , then maybe i 'm very junior in seniority in terms of where my loan is and maybe i get nothing . or i get a few pennies on the dollar here . so maybe i thought this was $ 1 billion and i have to write this down to $ 0.5 billion . so now i have two problems . i have this and i have this . and once again , this is a non liquid loan . bank a is in bankruptcy . and if i wanted to somehow get the value of this i have to wait for all of bank a 's assets to go into liquidation . and then whatever assets i get i would have to sell it . so this is kind of a frozen asset . so once again , i 'm stuck holding this non liquid asset . so now i have this non liquid asset that 's probably not worth what i thought it was , which was a loan to a . then i also have these cdos . and now , god forbid , let 's say that i had another loan to bank d. and now let 's say bank d goes bankrupt . and then i have another loan that 's bad on top of these cdos . but the cdos were the crux of the issue . that 's what caused the situation . if bank a could have only sold this cdo for $ 2 billion , it would n't have caused this chain reaction . and lehman brothers really was the thing that catalyzed this whole chain of events . and then you can imagine now bank c is worried because now bank b has all of these illiquid assets on top of these cdos and it starts to look bad . and you can imagine , now it 's even less likely that when a bank , let 's say that bank d is the next one to go into a dire situation , it 's even less likely that bank d can get a loan from a third bank . because all the banks are getting scared now . all the banks are saying , i 'm not going to loan money to anyone . if i can get any cash from anybody i 'm just going to keep it . so that when it 's my turn , when the market starts looking at me , i at least have a little cash . so everyone is frozen . everyone wants to collect their loans from everyone else and no one wants to give loans to anybody else . so that 's the situation we 're in . and that 's the difficulty that the fed is somehow trying to unwind . and i realized i 'm out of time again . i will confront that issue in the next video .
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and let 's say that someone else , just for fun , just to make this interesting -- i think you can extrapolate and think about how this gets complicated very fast . bank b has borrowed money from bank c. so bank c will have an asset here that says , no i lent money to bank b . fair enough .
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why does n't it come to bank c to claim the money bank c owes to a back ?
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i think we 're now ready to tackle the big picture and what has our government officials so worried right now . so what i 've done is , i 've just drawn the balance sheets for a bunch of banks . obviously , this is simplified . and i made all of their balance sheets look the same . all of these banks , each of these kind of represents the balance sheet of a bank . and just to explain it , the left-hand side of this balance sheet , so this column right here -- and maybe i can , at least for the first bank , mark it a little bit . so what i 'm squaring off in magenta , that 's the assets of that bank . what i 'm squaring off in blue , that 's the liabilities of the bank . and what i wrote here is , it has $ 4 billion of liabilities . its assets , i divided it between $ 3 billion of other assets and $ 2 billion of cdos . because we want to focus on the cdos , because that 's the crux of everything that 's going on . and we have $ 5 billion in assets , $ 4 billion of liabilities , so you have $ 1 billion in equity . so that 's what 's left there . so this is just another visual representation that liabilities plus equity is equal to assets . or assets minus liabilities is equal to equity . and i 've just copied and pasted this one balance sheet a bunch of times . i do n't know whether we 're going to use all those . but let 's just assume , for simplicity , that a ton of banks in the system have this identical balance sheet . obviously , they do n't have an identical balance sheet . but all of their balance sheets might have kind of similar properties . this is n't always the case , different banks have different exposures to cdos . some of them have a lot , some of them have a little bit . some of them are valuing them more conservatively than others . but just for the sake of simplicity , i 've just made all the banks in the situation where the book value of the cdos that they have on their balance sheets is the larger than their equity value . and i did that for a reason . because it leads to the issue of , are these banks facing just a liquidity issue or are they facing just a solvency issue ? if you believe that these are worth $ 3 billion , these assets , these liabilities are worth $ 4 billion , then the crux of whether it 's a liquidity or a solvency issue all falls down as to whether these are worth $ 2 billion or not . for example , if these are worth $ 2 billion , then you have $ 1 billion of equity . if these are worth $ 1.5 billion , well maybe they 're being a little optimistic here , but you 'll still have $ 0.5 billion of equity . so you 're still solvent . and in that situation , in theory , one is just if they do n't have the cash when some of their debt comes due , they should just be able to borrow some money and get past that hurdle . and then in the future maybe sell their assets and still have positive equity . however , if the true value of those cdos , and this is kind of a philosophical question , what 's the true value of anything ? and the best thing that we as humans have been to be able to come up with is a market . the market value tends to be the best representation of the true value of something . let 's say the true value of this is $ 1 billion or less , then we have a situation . for example , if these are worth nothing , then we only have $ 3 billion of assets , $ 4 billion of liabilities , we have negative equity . this company is worth nothing . and to lend this bank or this company any money would just be throwing good money after bad . because that money is just going to go into a black hole . because one of the people who this company owes money to is probably not going to see their money . and if you are the most junior person lending the money -- which means that when all the money is distributed if they go into bankruptcy , you 're the last person to see the money -- then you 're just throwing good money after bad . so that 's the issue . but i want you to see the big picture now . because if it was just an issue with one bank it would n't be a big deal . if it was just bear stearns or if it was just lehman brothers , not a big deal , let the greedy bankers go bankrupt . and they probably are doing just fine with the bonuses they 've collected after sourcing these cdos for the past eight years or five years or however long . but what i want to show you in this video is what people are talking about when they say systemic risk . so these $ 4 billion in liabilities , these are loans , maybe from other banks . in fact , probably from other banks . and those loans from other banks , those are assets of other banks . for example , let 's say this is bank a , this is bank b . maybe a billion of these are a loan from bank b . and if this is a loan from bank b , bank b would have an asset called loan to bank a . on bank b 's balance sheet we 're calling this a loan to bank a . this is one of its assets . and then one of its liabilities will be a loan from bank b . so how can i say this ? they took this money and they gave it to b. i 'm sorry , b had money , gave it to a in the form of a loan . and so that cash ended up here . and they got an asset called loan to bank a . and this is a liability , loan from bank b . and they might have taken that money and they might have lent it to bank c down here . i think you 're starting to see how this gets pretty hairy very fast . so let 's say that bank a , one of its $ 3 billion in assets , is a loan to bank c. and so on bank c 's balance sheet , it 'll say loan from bank a . or so we owe a $ 1 billion . and a says , c owes me $ 1 billion , and that 's all fine . and then you see that oh , we owe b $ 1 billion . and then we could keep doing this . or i could just even make this into a circle already . so maybe bank b has some money that it owes to someone else . and let 's say that someone else , just for fun , just to make this interesting -- i think you can extrapolate and think about how this gets complicated very fast . bank b has borrowed money from bank c. so bank c will have an asset here that says , no i lent money to bank b . fair enough . ok , so now we 're in an interesting situation . let 's say this loan , the loan from bank b to bank a comes due . and we 've studied this multiple times . and let 's say for whatever reason , all of these other loans , they 're not liquid . they 're not due yet . so bank a ca n't get rid of these loans . so let 's say this comes due , this is $ 4 billion . they ca n't sell any of this . so bank a has to come up with $ 1 billion somehow for bank b . so that 's the situation we 're dealing with . i 'm just going to say that they ca n't sell any of these assets . so it all comes down to the cdos . so there 's a couple of issues here . if you think it is just an issue of illiquidity , if these are $ 2 billion of assets , they 're really worth $ 2 billion , but bank a just ca n't sell them . because either there 's quote-unquote nobody willing to buy . although , i would argue if no-one is willing to buy something , then its true value is probably zero . but let 's just say bank a says no-one is willing to buy , we 're just illiquid , this is really worth $ 2 billion . so one situation is they could get a loan from someone . maybe the fed would be willing to take this as collateral . so they would give this as collateral to the fed . maybe the fed will give them a billion dollar loan . and then they can use that to pay bank b . let 's say that 's off the table because this is just smelly enough collateral that not even the fed , which we now realize is willing to do anything to support the markets , not even the fed is willing to give them a loan . or enough of a loan to pay off that loan . the other situation is maybe they can get an equity infusion from a sovereign wealth fund . and we covered that a couple of videos ago . where the sovereign wealth fund will essentially inject some cash . it 'll dilute the shares and then you know maybe we had 500 million shares before . now we 'll have 2 billion shares . so the sovereign wealth fund will take over roughly 80 % of the company . and in exchange for 80 % of the company , would give maybe $ 2 billion and then you could use that to pay off this loan . but let 's say that that 's not on the table anymore either . because the sovereign wealth funds have gotten burned so much . so what happens ? well we learned what happens . if you ca n't get a loan , a new loan , to replace this loan , or if you ca n't get an equity infusion from kind of a greater fool , what happens ? you go into bankruptcy . and this is what happened to lehman brothers . lehman brothers went into bankruptcy . no sovereign wealth fund , no one else bought the company . and i should probably do another video on that scenario . and they could n't get a loan . so they went bankrupt . i should call this company l actually . but i 'll call it company a for now . because i do n't want to impugn anyone . i actually do n't think lehman was any worse or better than any of the other players here . so when they go into bankruptcy , something very interesting happens . now , bank b , they were already worried about these cdos . these cdos were already an issue . and they were probably thinking , boy when when loan c comes due , i 'm going to be in trouble . or when loan d , or f , or whatever , i 'm going to be in trouble because i 'm going to be in that situation that i 'm essentially forcing bank a into right now . but now i have a new problem . this loan to bank a is n't getting paid off . and who knows ? bank a is going to go into bankruptcy . maybe in bankruptcy we realize that these are worth nothing . and if those are worth nothing , then maybe i 'm very junior in seniority in terms of where my loan is and maybe i get nothing . or i get a few pennies on the dollar here . so maybe i thought this was $ 1 billion and i have to write this down to $ 0.5 billion . so now i have two problems . i have this and i have this . and once again , this is a non liquid loan . bank a is in bankruptcy . and if i wanted to somehow get the value of this i have to wait for all of bank a 's assets to go into liquidation . and then whatever assets i get i would have to sell it . so this is kind of a frozen asset . so once again , i 'm stuck holding this non liquid asset . so now i have this non liquid asset that 's probably not worth what i thought it was , which was a loan to a . then i also have these cdos . and now , god forbid , let 's say that i had another loan to bank d. and now let 's say bank d goes bankrupt . and then i have another loan that 's bad on top of these cdos . but the cdos were the crux of the issue . that 's what caused the situation . if bank a could have only sold this cdo for $ 2 billion , it would n't have caused this chain reaction . and lehman brothers really was the thing that catalyzed this whole chain of events . and then you can imagine now bank c is worried because now bank b has all of these illiquid assets on top of these cdos and it starts to look bad . and you can imagine , now it 's even less likely that when a bank , let 's say that bank d is the next one to go into a dire situation , it 's even less likely that bank d can get a loan from a third bank . because all the banks are getting scared now . all the banks are saying , i 'm not going to loan money to anyone . if i can get any cash from anybody i 'm just going to keep it . so that when it 's my turn , when the market starts looking at me , i at least have a little cash . so everyone is frozen . everyone wants to collect their loans from everyone else and no one wants to give loans to anybody else . so that 's the situation we 're in . and that 's the difficulty that the fed is somehow trying to unwind . and i realized i 'm out of time again . i will confront that issue in the next video .
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maybe a billion of these are a loan from bank b . and if this is a loan from bank b , bank b would have an asset called loan to bank a . on bank b 's balance sheet we 're calling this a loan to bank a .
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a then would have $ 1b cash to repay bank b , would n't it ?
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i think we 're now ready to tackle the big picture and what has our government officials so worried right now . so what i 've done is , i 've just drawn the balance sheets for a bunch of banks . obviously , this is simplified . and i made all of their balance sheets look the same . all of these banks , each of these kind of represents the balance sheet of a bank . and just to explain it , the left-hand side of this balance sheet , so this column right here -- and maybe i can , at least for the first bank , mark it a little bit . so what i 'm squaring off in magenta , that 's the assets of that bank . what i 'm squaring off in blue , that 's the liabilities of the bank . and what i wrote here is , it has $ 4 billion of liabilities . its assets , i divided it between $ 3 billion of other assets and $ 2 billion of cdos . because we want to focus on the cdos , because that 's the crux of everything that 's going on . and we have $ 5 billion in assets , $ 4 billion of liabilities , so you have $ 1 billion in equity . so that 's what 's left there . so this is just another visual representation that liabilities plus equity is equal to assets . or assets minus liabilities is equal to equity . and i 've just copied and pasted this one balance sheet a bunch of times . i do n't know whether we 're going to use all those . but let 's just assume , for simplicity , that a ton of banks in the system have this identical balance sheet . obviously , they do n't have an identical balance sheet . but all of their balance sheets might have kind of similar properties . this is n't always the case , different banks have different exposures to cdos . some of them have a lot , some of them have a little bit . some of them are valuing them more conservatively than others . but just for the sake of simplicity , i 've just made all the banks in the situation where the book value of the cdos that they have on their balance sheets is the larger than their equity value . and i did that for a reason . because it leads to the issue of , are these banks facing just a liquidity issue or are they facing just a solvency issue ? if you believe that these are worth $ 3 billion , these assets , these liabilities are worth $ 4 billion , then the crux of whether it 's a liquidity or a solvency issue all falls down as to whether these are worth $ 2 billion or not . for example , if these are worth $ 2 billion , then you have $ 1 billion of equity . if these are worth $ 1.5 billion , well maybe they 're being a little optimistic here , but you 'll still have $ 0.5 billion of equity . so you 're still solvent . and in that situation , in theory , one is just if they do n't have the cash when some of their debt comes due , they should just be able to borrow some money and get past that hurdle . and then in the future maybe sell their assets and still have positive equity . however , if the true value of those cdos , and this is kind of a philosophical question , what 's the true value of anything ? and the best thing that we as humans have been to be able to come up with is a market . the market value tends to be the best representation of the true value of something . let 's say the true value of this is $ 1 billion or less , then we have a situation . for example , if these are worth nothing , then we only have $ 3 billion of assets , $ 4 billion of liabilities , we have negative equity . this company is worth nothing . and to lend this bank or this company any money would just be throwing good money after bad . because that money is just going to go into a black hole . because one of the people who this company owes money to is probably not going to see their money . and if you are the most junior person lending the money -- which means that when all the money is distributed if they go into bankruptcy , you 're the last person to see the money -- then you 're just throwing good money after bad . so that 's the issue . but i want you to see the big picture now . because if it was just an issue with one bank it would n't be a big deal . if it was just bear stearns or if it was just lehman brothers , not a big deal , let the greedy bankers go bankrupt . and they probably are doing just fine with the bonuses they 've collected after sourcing these cdos for the past eight years or five years or however long . but what i want to show you in this video is what people are talking about when they say systemic risk . so these $ 4 billion in liabilities , these are loans , maybe from other banks . in fact , probably from other banks . and those loans from other banks , those are assets of other banks . for example , let 's say this is bank a , this is bank b . maybe a billion of these are a loan from bank b . and if this is a loan from bank b , bank b would have an asset called loan to bank a . on bank b 's balance sheet we 're calling this a loan to bank a . this is one of its assets . and then one of its liabilities will be a loan from bank b . so how can i say this ? they took this money and they gave it to b. i 'm sorry , b had money , gave it to a in the form of a loan . and so that cash ended up here . and they got an asset called loan to bank a . and this is a liability , loan from bank b . and they might have taken that money and they might have lent it to bank c down here . i think you 're starting to see how this gets pretty hairy very fast . so let 's say that bank a , one of its $ 3 billion in assets , is a loan to bank c. and so on bank c 's balance sheet , it 'll say loan from bank a . or so we owe a $ 1 billion . and a says , c owes me $ 1 billion , and that 's all fine . and then you see that oh , we owe b $ 1 billion . and then we could keep doing this . or i could just even make this into a circle already . so maybe bank b has some money that it owes to someone else . and let 's say that someone else , just for fun , just to make this interesting -- i think you can extrapolate and think about how this gets complicated very fast . bank b has borrowed money from bank c. so bank c will have an asset here that says , no i lent money to bank b . fair enough . ok , so now we 're in an interesting situation . let 's say this loan , the loan from bank b to bank a comes due . and we 've studied this multiple times . and let 's say for whatever reason , all of these other loans , they 're not liquid . they 're not due yet . so bank a ca n't get rid of these loans . so let 's say this comes due , this is $ 4 billion . they ca n't sell any of this . so bank a has to come up with $ 1 billion somehow for bank b . so that 's the situation we 're dealing with . i 'm just going to say that they ca n't sell any of these assets . so it all comes down to the cdos . so there 's a couple of issues here . if you think it is just an issue of illiquidity , if these are $ 2 billion of assets , they 're really worth $ 2 billion , but bank a just ca n't sell them . because either there 's quote-unquote nobody willing to buy . although , i would argue if no-one is willing to buy something , then its true value is probably zero . but let 's just say bank a says no-one is willing to buy , we 're just illiquid , this is really worth $ 2 billion . so one situation is they could get a loan from someone . maybe the fed would be willing to take this as collateral . so they would give this as collateral to the fed . maybe the fed will give them a billion dollar loan . and then they can use that to pay bank b . let 's say that 's off the table because this is just smelly enough collateral that not even the fed , which we now realize is willing to do anything to support the markets , not even the fed is willing to give them a loan . or enough of a loan to pay off that loan . the other situation is maybe they can get an equity infusion from a sovereign wealth fund . and we covered that a couple of videos ago . where the sovereign wealth fund will essentially inject some cash . it 'll dilute the shares and then you know maybe we had 500 million shares before . now we 'll have 2 billion shares . so the sovereign wealth fund will take over roughly 80 % of the company . and in exchange for 80 % of the company , would give maybe $ 2 billion and then you could use that to pay off this loan . but let 's say that that 's not on the table anymore either . because the sovereign wealth funds have gotten burned so much . so what happens ? well we learned what happens . if you ca n't get a loan , a new loan , to replace this loan , or if you ca n't get an equity infusion from kind of a greater fool , what happens ? you go into bankruptcy . and this is what happened to lehman brothers . lehman brothers went into bankruptcy . no sovereign wealth fund , no one else bought the company . and i should probably do another video on that scenario . and they could n't get a loan . so they went bankrupt . i should call this company l actually . but i 'll call it company a for now . because i do n't want to impugn anyone . i actually do n't think lehman was any worse or better than any of the other players here . so when they go into bankruptcy , something very interesting happens . now , bank b , they were already worried about these cdos . these cdos were already an issue . and they were probably thinking , boy when when loan c comes due , i 'm going to be in trouble . or when loan d , or f , or whatever , i 'm going to be in trouble because i 'm going to be in that situation that i 'm essentially forcing bank a into right now . but now i have a new problem . this loan to bank a is n't getting paid off . and who knows ? bank a is going to go into bankruptcy . maybe in bankruptcy we realize that these are worth nothing . and if those are worth nothing , then maybe i 'm very junior in seniority in terms of where my loan is and maybe i get nothing . or i get a few pennies on the dollar here . so maybe i thought this was $ 1 billion and i have to write this down to $ 0.5 billion . so now i have two problems . i have this and i have this . and once again , this is a non liquid loan . bank a is in bankruptcy . and if i wanted to somehow get the value of this i have to wait for all of bank a 's assets to go into liquidation . and then whatever assets i get i would have to sell it . so this is kind of a frozen asset . so once again , i 'm stuck holding this non liquid asset . so now i have this non liquid asset that 's probably not worth what i thought it was , which was a loan to a . then i also have these cdos . and now , god forbid , let 's say that i had another loan to bank d. and now let 's say bank d goes bankrupt . and then i have another loan that 's bad on top of these cdos . but the cdos were the crux of the issue . that 's what caused the situation . if bank a could have only sold this cdo for $ 2 billion , it would n't have caused this chain reaction . and lehman brothers really was the thing that catalyzed this whole chain of events . and then you can imagine now bank c is worried because now bank b has all of these illiquid assets on top of these cdos and it starts to look bad . and you can imagine , now it 's even less likely that when a bank , let 's say that bank d is the next one to go into a dire situation , it 's even less likely that bank d can get a loan from a third bank . because all the banks are getting scared now . all the banks are saying , i 'm not going to loan money to anyone . if i can get any cash from anybody i 'm just going to keep it . so that when it 's my turn , when the market starts looking at me , i at least have a little cash . so everyone is frozen . everyone wants to collect their loans from everyone else and no one wants to give loans to anybody else . so that 's the situation we 're in . and that 's the difficulty that the fed is somehow trying to unwind . and i realized i 'm out of time again . i will confront that issue in the next video .
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in fact , probably from other banks . and those loans from other banks , those are assets of other banks . for example , let 's say this is bank a , this is bank b .
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also , i have quite a funny idea in my mind : if there is , say , an inter-banks loans , why do n't they cancel out each other 's debt ?
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i think we 're now ready to tackle the big picture and what has our government officials so worried right now . so what i 've done is , i 've just drawn the balance sheets for a bunch of banks . obviously , this is simplified . and i made all of their balance sheets look the same . all of these banks , each of these kind of represents the balance sheet of a bank . and just to explain it , the left-hand side of this balance sheet , so this column right here -- and maybe i can , at least for the first bank , mark it a little bit . so what i 'm squaring off in magenta , that 's the assets of that bank . what i 'm squaring off in blue , that 's the liabilities of the bank . and what i wrote here is , it has $ 4 billion of liabilities . its assets , i divided it between $ 3 billion of other assets and $ 2 billion of cdos . because we want to focus on the cdos , because that 's the crux of everything that 's going on . and we have $ 5 billion in assets , $ 4 billion of liabilities , so you have $ 1 billion in equity . so that 's what 's left there . so this is just another visual representation that liabilities plus equity is equal to assets . or assets minus liabilities is equal to equity . and i 've just copied and pasted this one balance sheet a bunch of times . i do n't know whether we 're going to use all those . but let 's just assume , for simplicity , that a ton of banks in the system have this identical balance sheet . obviously , they do n't have an identical balance sheet . but all of their balance sheets might have kind of similar properties . this is n't always the case , different banks have different exposures to cdos . some of them have a lot , some of them have a little bit . some of them are valuing them more conservatively than others . but just for the sake of simplicity , i 've just made all the banks in the situation where the book value of the cdos that they have on their balance sheets is the larger than their equity value . and i did that for a reason . because it leads to the issue of , are these banks facing just a liquidity issue or are they facing just a solvency issue ? if you believe that these are worth $ 3 billion , these assets , these liabilities are worth $ 4 billion , then the crux of whether it 's a liquidity or a solvency issue all falls down as to whether these are worth $ 2 billion or not . for example , if these are worth $ 2 billion , then you have $ 1 billion of equity . if these are worth $ 1.5 billion , well maybe they 're being a little optimistic here , but you 'll still have $ 0.5 billion of equity . so you 're still solvent . and in that situation , in theory , one is just if they do n't have the cash when some of their debt comes due , they should just be able to borrow some money and get past that hurdle . and then in the future maybe sell their assets and still have positive equity . however , if the true value of those cdos , and this is kind of a philosophical question , what 's the true value of anything ? and the best thing that we as humans have been to be able to come up with is a market . the market value tends to be the best representation of the true value of something . let 's say the true value of this is $ 1 billion or less , then we have a situation . for example , if these are worth nothing , then we only have $ 3 billion of assets , $ 4 billion of liabilities , we have negative equity . this company is worth nothing . and to lend this bank or this company any money would just be throwing good money after bad . because that money is just going to go into a black hole . because one of the people who this company owes money to is probably not going to see their money . and if you are the most junior person lending the money -- which means that when all the money is distributed if they go into bankruptcy , you 're the last person to see the money -- then you 're just throwing good money after bad . so that 's the issue . but i want you to see the big picture now . because if it was just an issue with one bank it would n't be a big deal . if it was just bear stearns or if it was just lehman brothers , not a big deal , let the greedy bankers go bankrupt . and they probably are doing just fine with the bonuses they 've collected after sourcing these cdos for the past eight years or five years or however long . but what i want to show you in this video is what people are talking about when they say systemic risk . so these $ 4 billion in liabilities , these are loans , maybe from other banks . in fact , probably from other banks . and those loans from other banks , those are assets of other banks . for example , let 's say this is bank a , this is bank b . maybe a billion of these are a loan from bank b . and if this is a loan from bank b , bank b would have an asset called loan to bank a . on bank b 's balance sheet we 're calling this a loan to bank a . this is one of its assets . and then one of its liabilities will be a loan from bank b . so how can i say this ? they took this money and they gave it to b. i 'm sorry , b had money , gave it to a in the form of a loan . and so that cash ended up here . and they got an asset called loan to bank a . and this is a liability , loan from bank b . and they might have taken that money and they might have lent it to bank c down here . i think you 're starting to see how this gets pretty hairy very fast . so let 's say that bank a , one of its $ 3 billion in assets , is a loan to bank c. and so on bank c 's balance sheet , it 'll say loan from bank a . or so we owe a $ 1 billion . and a says , c owes me $ 1 billion , and that 's all fine . and then you see that oh , we owe b $ 1 billion . and then we could keep doing this . or i could just even make this into a circle already . so maybe bank b has some money that it owes to someone else . and let 's say that someone else , just for fun , just to make this interesting -- i think you can extrapolate and think about how this gets complicated very fast . bank b has borrowed money from bank c. so bank c will have an asset here that says , no i lent money to bank b . fair enough . ok , so now we 're in an interesting situation . let 's say this loan , the loan from bank b to bank a comes due . and we 've studied this multiple times . and let 's say for whatever reason , all of these other loans , they 're not liquid . they 're not due yet . so bank a ca n't get rid of these loans . so let 's say this comes due , this is $ 4 billion . they ca n't sell any of this . so bank a has to come up with $ 1 billion somehow for bank b . so that 's the situation we 're dealing with . i 'm just going to say that they ca n't sell any of these assets . so it all comes down to the cdos . so there 's a couple of issues here . if you think it is just an issue of illiquidity , if these are $ 2 billion of assets , they 're really worth $ 2 billion , but bank a just ca n't sell them . because either there 's quote-unquote nobody willing to buy . although , i would argue if no-one is willing to buy something , then its true value is probably zero . but let 's just say bank a says no-one is willing to buy , we 're just illiquid , this is really worth $ 2 billion . so one situation is they could get a loan from someone . maybe the fed would be willing to take this as collateral . so they would give this as collateral to the fed . maybe the fed will give them a billion dollar loan . and then they can use that to pay bank b . let 's say that 's off the table because this is just smelly enough collateral that not even the fed , which we now realize is willing to do anything to support the markets , not even the fed is willing to give them a loan . or enough of a loan to pay off that loan . the other situation is maybe they can get an equity infusion from a sovereign wealth fund . and we covered that a couple of videos ago . where the sovereign wealth fund will essentially inject some cash . it 'll dilute the shares and then you know maybe we had 500 million shares before . now we 'll have 2 billion shares . so the sovereign wealth fund will take over roughly 80 % of the company . and in exchange for 80 % of the company , would give maybe $ 2 billion and then you could use that to pay off this loan . but let 's say that that 's not on the table anymore either . because the sovereign wealth funds have gotten burned so much . so what happens ? well we learned what happens . if you ca n't get a loan , a new loan , to replace this loan , or if you ca n't get an equity infusion from kind of a greater fool , what happens ? you go into bankruptcy . and this is what happened to lehman brothers . lehman brothers went into bankruptcy . no sovereign wealth fund , no one else bought the company . and i should probably do another video on that scenario . and they could n't get a loan . so they went bankrupt . i should call this company l actually . but i 'll call it company a for now . because i do n't want to impugn anyone . i actually do n't think lehman was any worse or better than any of the other players here . so when they go into bankruptcy , something very interesting happens . now , bank b , they were already worried about these cdos . these cdos were already an issue . and they were probably thinking , boy when when loan c comes due , i 'm going to be in trouble . or when loan d , or f , or whatever , i 'm going to be in trouble because i 'm going to be in that situation that i 'm essentially forcing bank a into right now . but now i have a new problem . this loan to bank a is n't getting paid off . and who knows ? bank a is going to go into bankruptcy . maybe in bankruptcy we realize that these are worth nothing . and if those are worth nothing , then maybe i 'm very junior in seniority in terms of where my loan is and maybe i get nothing . or i get a few pennies on the dollar here . so maybe i thought this was $ 1 billion and i have to write this down to $ 0.5 billion . so now i have two problems . i have this and i have this . and once again , this is a non liquid loan . bank a is in bankruptcy . and if i wanted to somehow get the value of this i have to wait for all of bank a 's assets to go into liquidation . and then whatever assets i get i would have to sell it . so this is kind of a frozen asset . so once again , i 'm stuck holding this non liquid asset . so now i have this non liquid asset that 's probably not worth what i thought it was , which was a loan to a . then i also have these cdos . and now , god forbid , let 's say that i had another loan to bank d. and now let 's say bank d goes bankrupt . and then i have another loan that 's bad on top of these cdos . but the cdos were the crux of the issue . that 's what caused the situation . if bank a could have only sold this cdo for $ 2 billion , it would n't have caused this chain reaction . and lehman brothers really was the thing that catalyzed this whole chain of events . and then you can imagine now bank c is worried because now bank b has all of these illiquid assets on top of these cdos and it starts to look bad . and you can imagine , now it 's even less likely that when a bank , let 's say that bank d is the next one to go into a dire situation , it 's even less likely that bank d can get a loan from a third bank . because all the banks are getting scared now . all the banks are saying , i 'm not going to loan money to anyone . if i can get any cash from anybody i 'm just going to keep it . so that when it 's my turn , when the market starts looking at me , i at least have a little cash . so everyone is frozen . everyone wants to collect their loans from everyone else and no one wants to give loans to anybody else . so that 's the situation we 're in . and that 's the difficulty that the fed is somehow trying to unwind . and i realized i 'm out of time again . i will confront that issue in the next video .
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so one situation is they could get a loan from someone . maybe the fed would be willing to take this as collateral . so they would give this as collateral to the fed .
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what was the reason and why banks were willing to take on that risk on their asset ?
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i think we 're now ready to tackle the big picture and what has our government officials so worried right now . so what i 've done is , i 've just drawn the balance sheets for a bunch of banks . obviously , this is simplified . and i made all of their balance sheets look the same . all of these banks , each of these kind of represents the balance sheet of a bank . and just to explain it , the left-hand side of this balance sheet , so this column right here -- and maybe i can , at least for the first bank , mark it a little bit . so what i 'm squaring off in magenta , that 's the assets of that bank . what i 'm squaring off in blue , that 's the liabilities of the bank . and what i wrote here is , it has $ 4 billion of liabilities . its assets , i divided it between $ 3 billion of other assets and $ 2 billion of cdos . because we want to focus on the cdos , because that 's the crux of everything that 's going on . and we have $ 5 billion in assets , $ 4 billion of liabilities , so you have $ 1 billion in equity . so that 's what 's left there . so this is just another visual representation that liabilities plus equity is equal to assets . or assets minus liabilities is equal to equity . and i 've just copied and pasted this one balance sheet a bunch of times . i do n't know whether we 're going to use all those . but let 's just assume , for simplicity , that a ton of banks in the system have this identical balance sheet . obviously , they do n't have an identical balance sheet . but all of their balance sheets might have kind of similar properties . this is n't always the case , different banks have different exposures to cdos . some of them have a lot , some of them have a little bit . some of them are valuing them more conservatively than others . but just for the sake of simplicity , i 've just made all the banks in the situation where the book value of the cdos that they have on their balance sheets is the larger than their equity value . and i did that for a reason . because it leads to the issue of , are these banks facing just a liquidity issue or are they facing just a solvency issue ? if you believe that these are worth $ 3 billion , these assets , these liabilities are worth $ 4 billion , then the crux of whether it 's a liquidity or a solvency issue all falls down as to whether these are worth $ 2 billion or not . for example , if these are worth $ 2 billion , then you have $ 1 billion of equity . if these are worth $ 1.5 billion , well maybe they 're being a little optimistic here , but you 'll still have $ 0.5 billion of equity . so you 're still solvent . and in that situation , in theory , one is just if they do n't have the cash when some of their debt comes due , they should just be able to borrow some money and get past that hurdle . and then in the future maybe sell their assets and still have positive equity . however , if the true value of those cdos , and this is kind of a philosophical question , what 's the true value of anything ? and the best thing that we as humans have been to be able to come up with is a market . the market value tends to be the best representation of the true value of something . let 's say the true value of this is $ 1 billion or less , then we have a situation . for example , if these are worth nothing , then we only have $ 3 billion of assets , $ 4 billion of liabilities , we have negative equity . this company is worth nothing . and to lend this bank or this company any money would just be throwing good money after bad . because that money is just going to go into a black hole . because one of the people who this company owes money to is probably not going to see their money . and if you are the most junior person lending the money -- which means that when all the money is distributed if they go into bankruptcy , you 're the last person to see the money -- then you 're just throwing good money after bad . so that 's the issue . but i want you to see the big picture now . because if it was just an issue with one bank it would n't be a big deal . if it was just bear stearns or if it was just lehman brothers , not a big deal , let the greedy bankers go bankrupt . and they probably are doing just fine with the bonuses they 've collected after sourcing these cdos for the past eight years or five years or however long . but what i want to show you in this video is what people are talking about when they say systemic risk . so these $ 4 billion in liabilities , these are loans , maybe from other banks . in fact , probably from other banks . and those loans from other banks , those are assets of other banks . for example , let 's say this is bank a , this is bank b . maybe a billion of these are a loan from bank b . and if this is a loan from bank b , bank b would have an asset called loan to bank a . on bank b 's balance sheet we 're calling this a loan to bank a . this is one of its assets . and then one of its liabilities will be a loan from bank b . so how can i say this ? they took this money and they gave it to b. i 'm sorry , b had money , gave it to a in the form of a loan . and so that cash ended up here . and they got an asset called loan to bank a . and this is a liability , loan from bank b . and they might have taken that money and they might have lent it to bank c down here . i think you 're starting to see how this gets pretty hairy very fast . so let 's say that bank a , one of its $ 3 billion in assets , is a loan to bank c. and so on bank c 's balance sheet , it 'll say loan from bank a . or so we owe a $ 1 billion . and a says , c owes me $ 1 billion , and that 's all fine . and then you see that oh , we owe b $ 1 billion . and then we could keep doing this . or i could just even make this into a circle already . so maybe bank b has some money that it owes to someone else . and let 's say that someone else , just for fun , just to make this interesting -- i think you can extrapolate and think about how this gets complicated very fast . bank b has borrowed money from bank c. so bank c will have an asset here that says , no i lent money to bank b . fair enough . ok , so now we 're in an interesting situation . let 's say this loan , the loan from bank b to bank a comes due . and we 've studied this multiple times . and let 's say for whatever reason , all of these other loans , they 're not liquid . they 're not due yet . so bank a ca n't get rid of these loans . so let 's say this comes due , this is $ 4 billion . they ca n't sell any of this . so bank a has to come up with $ 1 billion somehow for bank b . so that 's the situation we 're dealing with . i 'm just going to say that they ca n't sell any of these assets . so it all comes down to the cdos . so there 's a couple of issues here . if you think it is just an issue of illiquidity , if these are $ 2 billion of assets , they 're really worth $ 2 billion , but bank a just ca n't sell them . because either there 's quote-unquote nobody willing to buy . although , i would argue if no-one is willing to buy something , then its true value is probably zero . but let 's just say bank a says no-one is willing to buy , we 're just illiquid , this is really worth $ 2 billion . so one situation is they could get a loan from someone . maybe the fed would be willing to take this as collateral . so they would give this as collateral to the fed . maybe the fed will give them a billion dollar loan . and then they can use that to pay bank b . let 's say that 's off the table because this is just smelly enough collateral that not even the fed , which we now realize is willing to do anything to support the markets , not even the fed is willing to give them a loan . or enough of a loan to pay off that loan . the other situation is maybe they can get an equity infusion from a sovereign wealth fund . and we covered that a couple of videos ago . where the sovereign wealth fund will essentially inject some cash . it 'll dilute the shares and then you know maybe we had 500 million shares before . now we 'll have 2 billion shares . so the sovereign wealth fund will take over roughly 80 % of the company . and in exchange for 80 % of the company , would give maybe $ 2 billion and then you could use that to pay off this loan . but let 's say that that 's not on the table anymore either . because the sovereign wealth funds have gotten burned so much . so what happens ? well we learned what happens . if you ca n't get a loan , a new loan , to replace this loan , or if you ca n't get an equity infusion from kind of a greater fool , what happens ? you go into bankruptcy . and this is what happened to lehman brothers . lehman brothers went into bankruptcy . no sovereign wealth fund , no one else bought the company . and i should probably do another video on that scenario . and they could n't get a loan . so they went bankrupt . i should call this company l actually . but i 'll call it company a for now . because i do n't want to impugn anyone . i actually do n't think lehman was any worse or better than any of the other players here . so when they go into bankruptcy , something very interesting happens . now , bank b , they were already worried about these cdos . these cdos were already an issue . and they were probably thinking , boy when when loan c comes due , i 'm going to be in trouble . or when loan d , or f , or whatever , i 'm going to be in trouble because i 'm going to be in that situation that i 'm essentially forcing bank a into right now . but now i have a new problem . this loan to bank a is n't getting paid off . and who knows ? bank a is going to go into bankruptcy . maybe in bankruptcy we realize that these are worth nothing . and if those are worth nothing , then maybe i 'm very junior in seniority in terms of where my loan is and maybe i get nothing . or i get a few pennies on the dollar here . so maybe i thought this was $ 1 billion and i have to write this down to $ 0.5 billion . so now i have two problems . i have this and i have this . and once again , this is a non liquid loan . bank a is in bankruptcy . and if i wanted to somehow get the value of this i have to wait for all of bank a 's assets to go into liquidation . and then whatever assets i get i would have to sell it . so this is kind of a frozen asset . so once again , i 'm stuck holding this non liquid asset . so now i have this non liquid asset that 's probably not worth what i thought it was , which was a loan to a . then i also have these cdos . and now , god forbid , let 's say that i had another loan to bank d. and now let 's say bank d goes bankrupt . and then i have another loan that 's bad on top of these cdos . but the cdos were the crux of the issue . that 's what caused the situation . if bank a could have only sold this cdo for $ 2 billion , it would n't have caused this chain reaction . and lehman brothers really was the thing that catalyzed this whole chain of events . and then you can imagine now bank c is worried because now bank b has all of these illiquid assets on top of these cdos and it starts to look bad . and you can imagine , now it 's even less likely that when a bank , let 's say that bank d is the next one to go into a dire situation , it 's even less likely that bank d can get a loan from a third bank . because all the banks are getting scared now . all the banks are saying , i 'm not going to loan money to anyone . if i can get any cash from anybody i 'm just going to keep it . so that when it 's my turn , when the market starts looking at me , i at least have a little cash . so everyone is frozen . everyone wants to collect their loans from everyone else and no one wants to give loans to anybody else . so that 's the situation we 're in . and that 's the difficulty that the fed is somehow trying to unwind . and i realized i 'm out of time again . i will confront that issue in the next video .
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you go into bankruptcy . and this is what happened to lehman brothers . lehman brothers went into bankruptcy . no sovereign wealth fund , no one else bought the company .
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would have been better for the world economy if the fed had saved lehman brothers ?
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i think we 're now ready to tackle the big picture and what has our government officials so worried right now . so what i 've done is , i 've just drawn the balance sheets for a bunch of banks . obviously , this is simplified . and i made all of their balance sheets look the same . all of these banks , each of these kind of represents the balance sheet of a bank . and just to explain it , the left-hand side of this balance sheet , so this column right here -- and maybe i can , at least for the first bank , mark it a little bit . so what i 'm squaring off in magenta , that 's the assets of that bank . what i 'm squaring off in blue , that 's the liabilities of the bank . and what i wrote here is , it has $ 4 billion of liabilities . its assets , i divided it between $ 3 billion of other assets and $ 2 billion of cdos . because we want to focus on the cdos , because that 's the crux of everything that 's going on . and we have $ 5 billion in assets , $ 4 billion of liabilities , so you have $ 1 billion in equity . so that 's what 's left there . so this is just another visual representation that liabilities plus equity is equal to assets . or assets minus liabilities is equal to equity . and i 've just copied and pasted this one balance sheet a bunch of times . i do n't know whether we 're going to use all those . but let 's just assume , for simplicity , that a ton of banks in the system have this identical balance sheet . obviously , they do n't have an identical balance sheet . but all of their balance sheets might have kind of similar properties . this is n't always the case , different banks have different exposures to cdos . some of them have a lot , some of them have a little bit . some of them are valuing them more conservatively than others . but just for the sake of simplicity , i 've just made all the banks in the situation where the book value of the cdos that they have on their balance sheets is the larger than their equity value . and i did that for a reason . because it leads to the issue of , are these banks facing just a liquidity issue or are they facing just a solvency issue ? if you believe that these are worth $ 3 billion , these assets , these liabilities are worth $ 4 billion , then the crux of whether it 's a liquidity or a solvency issue all falls down as to whether these are worth $ 2 billion or not . for example , if these are worth $ 2 billion , then you have $ 1 billion of equity . if these are worth $ 1.5 billion , well maybe they 're being a little optimistic here , but you 'll still have $ 0.5 billion of equity . so you 're still solvent . and in that situation , in theory , one is just if they do n't have the cash when some of their debt comes due , they should just be able to borrow some money and get past that hurdle . and then in the future maybe sell their assets and still have positive equity . however , if the true value of those cdos , and this is kind of a philosophical question , what 's the true value of anything ? and the best thing that we as humans have been to be able to come up with is a market . the market value tends to be the best representation of the true value of something . let 's say the true value of this is $ 1 billion or less , then we have a situation . for example , if these are worth nothing , then we only have $ 3 billion of assets , $ 4 billion of liabilities , we have negative equity . this company is worth nothing . and to lend this bank or this company any money would just be throwing good money after bad . because that money is just going to go into a black hole . because one of the people who this company owes money to is probably not going to see their money . and if you are the most junior person lending the money -- which means that when all the money is distributed if they go into bankruptcy , you 're the last person to see the money -- then you 're just throwing good money after bad . so that 's the issue . but i want you to see the big picture now . because if it was just an issue with one bank it would n't be a big deal . if it was just bear stearns or if it was just lehman brothers , not a big deal , let the greedy bankers go bankrupt . and they probably are doing just fine with the bonuses they 've collected after sourcing these cdos for the past eight years or five years or however long . but what i want to show you in this video is what people are talking about when they say systemic risk . so these $ 4 billion in liabilities , these are loans , maybe from other banks . in fact , probably from other banks . and those loans from other banks , those are assets of other banks . for example , let 's say this is bank a , this is bank b . maybe a billion of these are a loan from bank b . and if this is a loan from bank b , bank b would have an asset called loan to bank a . on bank b 's balance sheet we 're calling this a loan to bank a . this is one of its assets . and then one of its liabilities will be a loan from bank b . so how can i say this ? they took this money and they gave it to b. i 'm sorry , b had money , gave it to a in the form of a loan . and so that cash ended up here . and they got an asset called loan to bank a . and this is a liability , loan from bank b . and they might have taken that money and they might have lent it to bank c down here . i think you 're starting to see how this gets pretty hairy very fast . so let 's say that bank a , one of its $ 3 billion in assets , is a loan to bank c. and so on bank c 's balance sheet , it 'll say loan from bank a . or so we owe a $ 1 billion . and a says , c owes me $ 1 billion , and that 's all fine . and then you see that oh , we owe b $ 1 billion . and then we could keep doing this . or i could just even make this into a circle already . so maybe bank b has some money that it owes to someone else . and let 's say that someone else , just for fun , just to make this interesting -- i think you can extrapolate and think about how this gets complicated very fast . bank b has borrowed money from bank c. so bank c will have an asset here that says , no i lent money to bank b . fair enough . ok , so now we 're in an interesting situation . let 's say this loan , the loan from bank b to bank a comes due . and we 've studied this multiple times . and let 's say for whatever reason , all of these other loans , they 're not liquid . they 're not due yet . so bank a ca n't get rid of these loans . so let 's say this comes due , this is $ 4 billion . they ca n't sell any of this . so bank a has to come up with $ 1 billion somehow for bank b . so that 's the situation we 're dealing with . i 'm just going to say that they ca n't sell any of these assets . so it all comes down to the cdos . so there 's a couple of issues here . if you think it is just an issue of illiquidity , if these are $ 2 billion of assets , they 're really worth $ 2 billion , but bank a just ca n't sell them . because either there 's quote-unquote nobody willing to buy . although , i would argue if no-one is willing to buy something , then its true value is probably zero . but let 's just say bank a says no-one is willing to buy , we 're just illiquid , this is really worth $ 2 billion . so one situation is they could get a loan from someone . maybe the fed would be willing to take this as collateral . so they would give this as collateral to the fed . maybe the fed will give them a billion dollar loan . and then they can use that to pay bank b . let 's say that 's off the table because this is just smelly enough collateral that not even the fed , which we now realize is willing to do anything to support the markets , not even the fed is willing to give them a loan . or enough of a loan to pay off that loan . the other situation is maybe they can get an equity infusion from a sovereign wealth fund . and we covered that a couple of videos ago . where the sovereign wealth fund will essentially inject some cash . it 'll dilute the shares and then you know maybe we had 500 million shares before . now we 'll have 2 billion shares . so the sovereign wealth fund will take over roughly 80 % of the company . and in exchange for 80 % of the company , would give maybe $ 2 billion and then you could use that to pay off this loan . but let 's say that that 's not on the table anymore either . because the sovereign wealth funds have gotten burned so much . so what happens ? well we learned what happens . if you ca n't get a loan , a new loan , to replace this loan , or if you ca n't get an equity infusion from kind of a greater fool , what happens ? you go into bankruptcy . and this is what happened to lehman brothers . lehman brothers went into bankruptcy . no sovereign wealth fund , no one else bought the company . and i should probably do another video on that scenario . and they could n't get a loan . so they went bankrupt . i should call this company l actually . but i 'll call it company a for now . because i do n't want to impugn anyone . i actually do n't think lehman was any worse or better than any of the other players here . so when they go into bankruptcy , something very interesting happens . now , bank b , they were already worried about these cdos . these cdos were already an issue . and they were probably thinking , boy when when loan c comes due , i 'm going to be in trouble . or when loan d , or f , or whatever , i 'm going to be in trouble because i 'm going to be in that situation that i 'm essentially forcing bank a into right now . but now i have a new problem . this loan to bank a is n't getting paid off . and who knows ? bank a is going to go into bankruptcy . maybe in bankruptcy we realize that these are worth nothing . and if those are worth nothing , then maybe i 'm very junior in seniority in terms of where my loan is and maybe i get nothing . or i get a few pennies on the dollar here . so maybe i thought this was $ 1 billion and i have to write this down to $ 0.5 billion . so now i have two problems . i have this and i have this . and once again , this is a non liquid loan . bank a is in bankruptcy . and if i wanted to somehow get the value of this i have to wait for all of bank a 's assets to go into liquidation . and then whatever assets i get i would have to sell it . so this is kind of a frozen asset . so once again , i 'm stuck holding this non liquid asset . so now i have this non liquid asset that 's probably not worth what i thought it was , which was a loan to a . then i also have these cdos . and now , god forbid , let 's say that i had another loan to bank d. and now let 's say bank d goes bankrupt . and then i have another loan that 's bad on top of these cdos . but the cdos were the crux of the issue . that 's what caused the situation . if bank a could have only sold this cdo for $ 2 billion , it would n't have caused this chain reaction . and lehman brothers really was the thing that catalyzed this whole chain of events . and then you can imagine now bank c is worried because now bank b has all of these illiquid assets on top of these cdos and it starts to look bad . and you can imagine , now it 's even less likely that when a bank , let 's say that bank d is the next one to go into a dire situation , it 's even less likely that bank d can get a loan from a third bank . because all the banks are getting scared now . all the banks are saying , i 'm not going to loan money to anyone . if i can get any cash from anybody i 'm just going to keep it . so that when it 's my turn , when the market starts looking at me , i at least have a little cash . so everyone is frozen . everyone wants to collect their loans from everyone else and no one wants to give loans to anybody else . so that 's the situation we 're in . and that 's the difficulty that the fed is somehow trying to unwind . and i realized i 'm out of time again . i will confront that issue in the next video .
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and you can imagine , now it 's even less likely that when a bank , let 's say that bank d is the next one to go into a dire situation , it 's even less likely that bank d can get a loan from a third bank . because all the banks are getting scared now . all the banks are saying , i 'm not going to loan money to anyone . if i can get any cash from anybody i 'm just going to keep it .
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are banks ever aware of who is loaning whom money ?
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i think we 're now ready to tackle the big picture and what has our government officials so worried right now . so what i 've done is , i 've just drawn the balance sheets for a bunch of banks . obviously , this is simplified . and i made all of their balance sheets look the same . all of these banks , each of these kind of represents the balance sheet of a bank . and just to explain it , the left-hand side of this balance sheet , so this column right here -- and maybe i can , at least for the first bank , mark it a little bit . so what i 'm squaring off in magenta , that 's the assets of that bank . what i 'm squaring off in blue , that 's the liabilities of the bank . and what i wrote here is , it has $ 4 billion of liabilities . its assets , i divided it between $ 3 billion of other assets and $ 2 billion of cdos . because we want to focus on the cdos , because that 's the crux of everything that 's going on . and we have $ 5 billion in assets , $ 4 billion of liabilities , so you have $ 1 billion in equity . so that 's what 's left there . so this is just another visual representation that liabilities plus equity is equal to assets . or assets minus liabilities is equal to equity . and i 've just copied and pasted this one balance sheet a bunch of times . i do n't know whether we 're going to use all those . but let 's just assume , for simplicity , that a ton of banks in the system have this identical balance sheet . obviously , they do n't have an identical balance sheet . but all of their balance sheets might have kind of similar properties . this is n't always the case , different banks have different exposures to cdos . some of them have a lot , some of them have a little bit . some of them are valuing them more conservatively than others . but just for the sake of simplicity , i 've just made all the banks in the situation where the book value of the cdos that they have on their balance sheets is the larger than their equity value . and i did that for a reason . because it leads to the issue of , are these banks facing just a liquidity issue or are they facing just a solvency issue ? if you believe that these are worth $ 3 billion , these assets , these liabilities are worth $ 4 billion , then the crux of whether it 's a liquidity or a solvency issue all falls down as to whether these are worth $ 2 billion or not . for example , if these are worth $ 2 billion , then you have $ 1 billion of equity . if these are worth $ 1.5 billion , well maybe they 're being a little optimistic here , but you 'll still have $ 0.5 billion of equity . so you 're still solvent . and in that situation , in theory , one is just if they do n't have the cash when some of their debt comes due , they should just be able to borrow some money and get past that hurdle . and then in the future maybe sell their assets and still have positive equity . however , if the true value of those cdos , and this is kind of a philosophical question , what 's the true value of anything ? and the best thing that we as humans have been to be able to come up with is a market . the market value tends to be the best representation of the true value of something . let 's say the true value of this is $ 1 billion or less , then we have a situation . for example , if these are worth nothing , then we only have $ 3 billion of assets , $ 4 billion of liabilities , we have negative equity . this company is worth nothing . and to lend this bank or this company any money would just be throwing good money after bad . because that money is just going to go into a black hole . because one of the people who this company owes money to is probably not going to see their money . and if you are the most junior person lending the money -- which means that when all the money is distributed if they go into bankruptcy , you 're the last person to see the money -- then you 're just throwing good money after bad . so that 's the issue . but i want you to see the big picture now . because if it was just an issue with one bank it would n't be a big deal . if it was just bear stearns or if it was just lehman brothers , not a big deal , let the greedy bankers go bankrupt . and they probably are doing just fine with the bonuses they 've collected after sourcing these cdos for the past eight years or five years or however long . but what i want to show you in this video is what people are talking about when they say systemic risk . so these $ 4 billion in liabilities , these are loans , maybe from other banks . in fact , probably from other banks . and those loans from other banks , those are assets of other banks . for example , let 's say this is bank a , this is bank b . maybe a billion of these are a loan from bank b . and if this is a loan from bank b , bank b would have an asset called loan to bank a . on bank b 's balance sheet we 're calling this a loan to bank a . this is one of its assets . and then one of its liabilities will be a loan from bank b . so how can i say this ? they took this money and they gave it to b. i 'm sorry , b had money , gave it to a in the form of a loan . and so that cash ended up here . and they got an asset called loan to bank a . and this is a liability , loan from bank b . and they might have taken that money and they might have lent it to bank c down here . i think you 're starting to see how this gets pretty hairy very fast . so let 's say that bank a , one of its $ 3 billion in assets , is a loan to bank c. and so on bank c 's balance sheet , it 'll say loan from bank a . or so we owe a $ 1 billion . and a says , c owes me $ 1 billion , and that 's all fine . and then you see that oh , we owe b $ 1 billion . and then we could keep doing this . or i could just even make this into a circle already . so maybe bank b has some money that it owes to someone else . and let 's say that someone else , just for fun , just to make this interesting -- i think you can extrapolate and think about how this gets complicated very fast . bank b has borrowed money from bank c. so bank c will have an asset here that says , no i lent money to bank b . fair enough . ok , so now we 're in an interesting situation . let 's say this loan , the loan from bank b to bank a comes due . and we 've studied this multiple times . and let 's say for whatever reason , all of these other loans , they 're not liquid . they 're not due yet . so bank a ca n't get rid of these loans . so let 's say this comes due , this is $ 4 billion . they ca n't sell any of this . so bank a has to come up with $ 1 billion somehow for bank b . so that 's the situation we 're dealing with . i 'm just going to say that they ca n't sell any of these assets . so it all comes down to the cdos . so there 's a couple of issues here . if you think it is just an issue of illiquidity , if these are $ 2 billion of assets , they 're really worth $ 2 billion , but bank a just ca n't sell them . because either there 's quote-unquote nobody willing to buy . although , i would argue if no-one is willing to buy something , then its true value is probably zero . but let 's just say bank a says no-one is willing to buy , we 're just illiquid , this is really worth $ 2 billion . so one situation is they could get a loan from someone . maybe the fed would be willing to take this as collateral . so they would give this as collateral to the fed . maybe the fed will give them a billion dollar loan . and then they can use that to pay bank b . let 's say that 's off the table because this is just smelly enough collateral that not even the fed , which we now realize is willing to do anything to support the markets , not even the fed is willing to give them a loan . or enough of a loan to pay off that loan . the other situation is maybe they can get an equity infusion from a sovereign wealth fund . and we covered that a couple of videos ago . where the sovereign wealth fund will essentially inject some cash . it 'll dilute the shares and then you know maybe we had 500 million shares before . now we 'll have 2 billion shares . so the sovereign wealth fund will take over roughly 80 % of the company . and in exchange for 80 % of the company , would give maybe $ 2 billion and then you could use that to pay off this loan . but let 's say that that 's not on the table anymore either . because the sovereign wealth funds have gotten burned so much . so what happens ? well we learned what happens . if you ca n't get a loan , a new loan , to replace this loan , or if you ca n't get an equity infusion from kind of a greater fool , what happens ? you go into bankruptcy . and this is what happened to lehman brothers . lehman brothers went into bankruptcy . no sovereign wealth fund , no one else bought the company . and i should probably do another video on that scenario . and they could n't get a loan . so they went bankrupt . i should call this company l actually . but i 'll call it company a for now . because i do n't want to impugn anyone . i actually do n't think lehman was any worse or better than any of the other players here . so when they go into bankruptcy , something very interesting happens . now , bank b , they were already worried about these cdos . these cdos were already an issue . and they were probably thinking , boy when when loan c comes due , i 'm going to be in trouble . or when loan d , or f , or whatever , i 'm going to be in trouble because i 'm going to be in that situation that i 'm essentially forcing bank a into right now . but now i have a new problem . this loan to bank a is n't getting paid off . and who knows ? bank a is going to go into bankruptcy . maybe in bankruptcy we realize that these are worth nothing . and if those are worth nothing , then maybe i 'm very junior in seniority in terms of where my loan is and maybe i get nothing . or i get a few pennies on the dollar here . so maybe i thought this was $ 1 billion and i have to write this down to $ 0.5 billion . so now i have two problems . i have this and i have this . and once again , this is a non liquid loan . bank a is in bankruptcy . and if i wanted to somehow get the value of this i have to wait for all of bank a 's assets to go into liquidation . and then whatever assets i get i would have to sell it . so this is kind of a frozen asset . so once again , i 'm stuck holding this non liquid asset . so now i have this non liquid asset that 's probably not worth what i thought it was , which was a loan to a . then i also have these cdos . and now , god forbid , let 's say that i had another loan to bank d. and now let 's say bank d goes bankrupt . and then i have another loan that 's bad on top of these cdos . but the cdos were the crux of the issue . that 's what caused the situation . if bank a could have only sold this cdo for $ 2 billion , it would n't have caused this chain reaction . and lehman brothers really was the thing that catalyzed this whole chain of events . and then you can imagine now bank c is worried because now bank b has all of these illiquid assets on top of these cdos and it starts to look bad . and you can imagine , now it 's even less likely that when a bank , let 's say that bank d is the next one to go into a dire situation , it 's even less likely that bank d can get a loan from a third bank . because all the banks are getting scared now . all the banks are saying , i 'm not going to loan money to anyone . if i can get any cash from anybody i 'm just going to keep it . so that when it 's my turn , when the market starts looking at me , i at least have a little cash . so everyone is frozen . everyone wants to collect their loans from everyone else and no one wants to give loans to anybody else . so that 's the situation we 're in . and that 's the difficulty that the fed is somehow trying to unwind . and i realized i 'm out of time again . i will confront that issue in the next video .
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and let 's say that someone else , just for fun , just to make this interesting -- i think you can extrapolate and think about how this gets complicated very fast . bank b has borrowed money from bank c. so bank c will have an asset here that says , no i lent money to bank b . fair enough .
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like would bank b know that bank a was loaning money to c ?
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i think we 're now ready to tackle the big picture and what has our government officials so worried right now . so what i 've done is , i 've just drawn the balance sheets for a bunch of banks . obviously , this is simplified . and i made all of their balance sheets look the same . all of these banks , each of these kind of represents the balance sheet of a bank . and just to explain it , the left-hand side of this balance sheet , so this column right here -- and maybe i can , at least for the first bank , mark it a little bit . so what i 'm squaring off in magenta , that 's the assets of that bank . what i 'm squaring off in blue , that 's the liabilities of the bank . and what i wrote here is , it has $ 4 billion of liabilities . its assets , i divided it between $ 3 billion of other assets and $ 2 billion of cdos . because we want to focus on the cdos , because that 's the crux of everything that 's going on . and we have $ 5 billion in assets , $ 4 billion of liabilities , so you have $ 1 billion in equity . so that 's what 's left there . so this is just another visual representation that liabilities plus equity is equal to assets . or assets minus liabilities is equal to equity . and i 've just copied and pasted this one balance sheet a bunch of times . i do n't know whether we 're going to use all those . but let 's just assume , for simplicity , that a ton of banks in the system have this identical balance sheet . obviously , they do n't have an identical balance sheet . but all of their balance sheets might have kind of similar properties . this is n't always the case , different banks have different exposures to cdos . some of them have a lot , some of them have a little bit . some of them are valuing them more conservatively than others . but just for the sake of simplicity , i 've just made all the banks in the situation where the book value of the cdos that they have on their balance sheets is the larger than their equity value . and i did that for a reason . because it leads to the issue of , are these banks facing just a liquidity issue or are they facing just a solvency issue ? if you believe that these are worth $ 3 billion , these assets , these liabilities are worth $ 4 billion , then the crux of whether it 's a liquidity or a solvency issue all falls down as to whether these are worth $ 2 billion or not . for example , if these are worth $ 2 billion , then you have $ 1 billion of equity . if these are worth $ 1.5 billion , well maybe they 're being a little optimistic here , but you 'll still have $ 0.5 billion of equity . so you 're still solvent . and in that situation , in theory , one is just if they do n't have the cash when some of their debt comes due , they should just be able to borrow some money and get past that hurdle . and then in the future maybe sell their assets and still have positive equity . however , if the true value of those cdos , and this is kind of a philosophical question , what 's the true value of anything ? and the best thing that we as humans have been to be able to come up with is a market . the market value tends to be the best representation of the true value of something . let 's say the true value of this is $ 1 billion or less , then we have a situation . for example , if these are worth nothing , then we only have $ 3 billion of assets , $ 4 billion of liabilities , we have negative equity . this company is worth nothing . and to lend this bank or this company any money would just be throwing good money after bad . because that money is just going to go into a black hole . because one of the people who this company owes money to is probably not going to see their money . and if you are the most junior person lending the money -- which means that when all the money is distributed if they go into bankruptcy , you 're the last person to see the money -- then you 're just throwing good money after bad . so that 's the issue . but i want you to see the big picture now . because if it was just an issue with one bank it would n't be a big deal . if it was just bear stearns or if it was just lehman brothers , not a big deal , let the greedy bankers go bankrupt . and they probably are doing just fine with the bonuses they 've collected after sourcing these cdos for the past eight years or five years or however long . but what i want to show you in this video is what people are talking about when they say systemic risk . so these $ 4 billion in liabilities , these are loans , maybe from other banks . in fact , probably from other banks . and those loans from other banks , those are assets of other banks . for example , let 's say this is bank a , this is bank b . maybe a billion of these are a loan from bank b . and if this is a loan from bank b , bank b would have an asset called loan to bank a . on bank b 's balance sheet we 're calling this a loan to bank a . this is one of its assets . and then one of its liabilities will be a loan from bank b . so how can i say this ? they took this money and they gave it to b. i 'm sorry , b had money , gave it to a in the form of a loan . and so that cash ended up here . and they got an asset called loan to bank a . and this is a liability , loan from bank b . and they might have taken that money and they might have lent it to bank c down here . i think you 're starting to see how this gets pretty hairy very fast . so let 's say that bank a , one of its $ 3 billion in assets , is a loan to bank c. and so on bank c 's balance sheet , it 'll say loan from bank a . or so we owe a $ 1 billion . and a says , c owes me $ 1 billion , and that 's all fine . and then you see that oh , we owe b $ 1 billion . and then we could keep doing this . or i could just even make this into a circle already . so maybe bank b has some money that it owes to someone else . and let 's say that someone else , just for fun , just to make this interesting -- i think you can extrapolate and think about how this gets complicated very fast . bank b has borrowed money from bank c. so bank c will have an asset here that says , no i lent money to bank b . fair enough . ok , so now we 're in an interesting situation . let 's say this loan , the loan from bank b to bank a comes due . and we 've studied this multiple times . and let 's say for whatever reason , all of these other loans , they 're not liquid . they 're not due yet . so bank a ca n't get rid of these loans . so let 's say this comes due , this is $ 4 billion . they ca n't sell any of this . so bank a has to come up with $ 1 billion somehow for bank b . so that 's the situation we 're dealing with . i 'm just going to say that they ca n't sell any of these assets . so it all comes down to the cdos . so there 's a couple of issues here . if you think it is just an issue of illiquidity , if these are $ 2 billion of assets , they 're really worth $ 2 billion , but bank a just ca n't sell them . because either there 's quote-unquote nobody willing to buy . although , i would argue if no-one is willing to buy something , then its true value is probably zero . but let 's just say bank a says no-one is willing to buy , we 're just illiquid , this is really worth $ 2 billion . so one situation is they could get a loan from someone . maybe the fed would be willing to take this as collateral . so they would give this as collateral to the fed . maybe the fed will give them a billion dollar loan . and then they can use that to pay bank b . let 's say that 's off the table because this is just smelly enough collateral that not even the fed , which we now realize is willing to do anything to support the markets , not even the fed is willing to give them a loan . or enough of a loan to pay off that loan . the other situation is maybe they can get an equity infusion from a sovereign wealth fund . and we covered that a couple of videos ago . where the sovereign wealth fund will essentially inject some cash . it 'll dilute the shares and then you know maybe we had 500 million shares before . now we 'll have 2 billion shares . so the sovereign wealth fund will take over roughly 80 % of the company . and in exchange for 80 % of the company , would give maybe $ 2 billion and then you could use that to pay off this loan . but let 's say that that 's not on the table anymore either . because the sovereign wealth funds have gotten burned so much . so what happens ? well we learned what happens . if you ca n't get a loan , a new loan , to replace this loan , or if you ca n't get an equity infusion from kind of a greater fool , what happens ? you go into bankruptcy . and this is what happened to lehman brothers . lehman brothers went into bankruptcy . no sovereign wealth fund , no one else bought the company . and i should probably do another video on that scenario . and they could n't get a loan . so they went bankrupt . i should call this company l actually . but i 'll call it company a for now . because i do n't want to impugn anyone . i actually do n't think lehman was any worse or better than any of the other players here . so when they go into bankruptcy , something very interesting happens . now , bank b , they were already worried about these cdos . these cdos were already an issue . and they were probably thinking , boy when when loan c comes due , i 'm going to be in trouble . or when loan d , or f , or whatever , i 'm going to be in trouble because i 'm going to be in that situation that i 'm essentially forcing bank a into right now . but now i have a new problem . this loan to bank a is n't getting paid off . and who knows ? bank a is going to go into bankruptcy . maybe in bankruptcy we realize that these are worth nothing . and if those are worth nothing , then maybe i 'm very junior in seniority in terms of where my loan is and maybe i get nothing . or i get a few pennies on the dollar here . so maybe i thought this was $ 1 billion and i have to write this down to $ 0.5 billion . so now i have two problems . i have this and i have this . and once again , this is a non liquid loan . bank a is in bankruptcy . and if i wanted to somehow get the value of this i have to wait for all of bank a 's assets to go into liquidation . and then whatever assets i get i would have to sell it . so this is kind of a frozen asset . so once again , i 'm stuck holding this non liquid asset . so now i have this non liquid asset that 's probably not worth what i thought it was , which was a loan to a . then i also have these cdos . and now , god forbid , let 's say that i had another loan to bank d. and now let 's say bank d goes bankrupt . and then i have another loan that 's bad on top of these cdos . but the cdos were the crux of the issue . that 's what caused the situation . if bank a could have only sold this cdo for $ 2 billion , it would n't have caused this chain reaction . and lehman brothers really was the thing that catalyzed this whole chain of events . and then you can imagine now bank c is worried because now bank b has all of these illiquid assets on top of these cdos and it starts to look bad . and you can imagine , now it 's even less likely that when a bank , let 's say that bank d is the next one to go into a dire situation , it 's even less likely that bank d can get a loan from a third bank . because all the banks are getting scared now . all the banks are saying , i 'm not going to loan money to anyone . if i can get any cash from anybody i 'm just going to keep it . so that when it 's my turn , when the market starts looking at me , i at least have a little cash . so everyone is frozen . everyone wants to collect their loans from everyone else and no one wants to give loans to anybody else . so that 's the situation we 're in . and that 's the difficulty that the fed is somehow trying to unwind . and i realized i 'm out of time again . i will confront that issue in the next video .
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so they would give this as collateral to the fed . maybe the fed will give them a billion dollar loan . and then they can use that to pay bank b .
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when sal says , that the fed will give them a billion dollar loan , is he referring to the federal reserve , composed of member banks itself , or the federal government ?
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i think we 're now ready to tackle the big picture and what has our government officials so worried right now . so what i 've done is , i 've just drawn the balance sheets for a bunch of banks . obviously , this is simplified . and i made all of their balance sheets look the same . all of these banks , each of these kind of represents the balance sheet of a bank . and just to explain it , the left-hand side of this balance sheet , so this column right here -- and maybe i can , at least for the first bank , mark it a little bit . so what i 'm squaring off in magenta , that 's the assets of that bank . what i 'm squaring off in blue , that 's the liabilities of the bank . and what i wrote here is , it has $ 4 billion of liabilities . its assets , i divided it between $ 3 billion of other assets and $ 2 billion of cdos . because we want to focus on the cdos , because that 's the crux of everything that 's going on . and we have $ 5 billion in assets , $ 4 billion of liabilities , so you have $ 1 billion in equity . so that 's what 's left there . so this is just another visual representation that liabilities plus equity is equal to assets . or assets minus liabilities is equal to equity . and i 've just copied and pasted this one balance sheet a bunch of times . i do n't know whether we 're going to use all those . but let 's just assume , for simplicity , that a ton of banks in the system have this identical balance sheet . obviously , they do n't have an identical balance sheet . but all of their balance sheets might have kind of similar properties . this is n't always the case , different banks have different exposures to cdos . some of them have a lot , some of them have a little bit . some of them are valuing them more conservatively than others . but just for the sake of simplicity , i 've just made all the banks in the situation where the book value of the cdos that they have on their balance sheets is the larger than their equity value . and i did that for a reason . because it leads to the issue of , are these banks facing just a liquidity issue or are they facing just a solvency issue ? if you believe that these are worth $ 3 billion , these assets , these liabilities are worth $ 4 billion , then the crux of whether it 's a liquidity or a solvency issue all falls down as to whether these are worth $ 2 billion or not . for example , if these are worth $ 2 billion , then you have $ 1 billion of equity . if these are worth $ 1.5 billion , well maybe they 're being a little optimistic here , but you 'll still have $ 0.5 billion of equity . so you 're still solvent . and in that situation , in theory , one is just if they do n't have the cash when some of their debt comes due , they should just be able to borrow some money and get past that hurdle . and then in the future maybe sell their assets and still have positive equity . however , if the true value of those cdos , and this is kind of a philosophical question , what 's the true value of anything ? and the best thing that we as humans have been to be able to come up with is a market . the market value tends to be the best representation of the true value of something . let 's say the true value of this is $ 1 billion or less , then we have a situation . for example , if these are worth nothing , then we only have $ 3 billion of assets , $ 4 billion of liabilities , we have negative equity . this company is worth nothing . and to lend this bank or this company any money would just be throwing good money after bad . because that money is just going to go into a black hole . because one of the people who this company owes money to is probably not going to see their money . and if you are the most junior person lending the money -- which means that when all the money is distributed if they go into bankruptcy , you 're the last person to see the money -- then you 're just throwing good money after bad . so that 's the issue . but i want you to see the big picture now . because if it was just an issue with one bank it would n't be a big deal . if it was just bear stearns or if it was just lehman brothers , not a big deal , let the greedy bankers go bankrupt . and they probably are doing just fine with the bonuses they 've collected after sourcing these cdos for the past eight years or five years or however long . but what i want to show you in this video is what people are talking about when they say systemic risk . so these $ 4 billion in liabilities , these are loans , maybe from other banks . in fact , probably from other banks . and those loans from other banks , those are assets of other banks . for example , let 's say this is bank a , this is bank b . maybe a billion of these are a loan from bank b . and if this is a loan from bank b , bank b would have an asset called loan to bank a . on bank b 's balance sheet we 're calling this a loan to bank a . this is one of its assets . and then one of its liabilities will be a loan from bank b . so how can i say this ? they took this money and they gave it to b. i 'm sorry , b had money , gave it to a in the form of a loan . and so that cash ended up here . and they got an asset called loan to bank a . and this is a liability , loan from bank b . and they might have taken that money and they might have lent it to bank c down here . i think you 're starting to see how this gets pretty hairy very fast . so let 's say that bank a , one of its $ 3 billion in assets , is a loan to bank c. and so on bank c 's balance sheet , it 'll say loan from bank a . or so we owe a $ 1 billion . and a says , c owes me $ 1 billion , and that 's all fine . and then you see that oh , we owe b $ 1 billion . and then we could keep doing this . or i could just even make this into a circle already . so maybe bank b has some money that it owes to someone else . and let 's say that someone else , just for fun , just to make this interesting -- i think you can extrapolate and think about how this gets complicated very fast . bank b has borrowed money from bank c. so bank c will have an asset here that says , no i lent money to bank b . fair enough . ok , so now we 're in an interesting situation . let 's say this loan , the loan from bank b to bank a comes due . and we 've studied this multiple times . and let 's say for whatever reason , all of these other loans , they 're not liquid . they 're not due yet . so bank a ca n't get rid of these loans . so let 's say this comes due , this is $ 4 billion . they ca n't sell any of this . so bank a has to come up with $ 1 billion somehow for bank b . so that 's the situation we 're dealing with . i 'm just going to say that they ca n't sell any of these assets . so it all comes down to the cdos . so there 's a couple of issues here . if you think it is just an issue of illiquidity , if these are $ 2 billion of assets , they 're really worth $ 2 billion , but bank a just ca n't sell them . because either there 's quote-unquote nobody willing to buy . although , i would argue if no-one is willing to buy something , then its true value is probably zero . but let 's just say bank a says no-one is willing to buy , we 're just illiquid , this is really worth $ 2 billion . so one situation is they could get a loan from someone . maybe the fed would be willing to take this as collateral . so they would give this as collateral to the fed . maybe the fed will give them a billion dollar loan . and then they can use that to pay bank b . let 's say that 's off the table because this is just smelly enough collateral that not even the fed , which we now realize is willing to do anything to support the markets , not even the fed is willing to give them a loan . or enough of a loan to pay off that loan . the other situation is maybe they can get an equity infusion from a sovereign wealth fund . and we covered that a couple of videos ago . where the sovereign wealth fund will essentially inject some cash . it 'll dilute the shares and then you know maybe we had 500 million shares before . now we 'll have 2 billion shares . so the sovereign wealth fund will take over roughly 80 % of the company . and in exchange for 80 % of the company , would give maybe $ 2 billion and then you could use that to pay off this loan . but let 's say that that 's not on the table anymore either . because the sovereign wealth funds have gotten burned so much . so what happens ? well we learned what happens . if you ca n't get a loan , a new loan , to replace this loan , or if you ca n't get an equity infusion from kind of a greater fool , what happens ? you go into bankruptcy . and this is what happened to lehman brothers . lehman brothers went into bankruptcy . no sovereign wealth fund , no one else bought the company . and i should probably do another video on that scenario . and they could n't get a loan . so they went bankrupt . i should call this company l actually . but i 'll call it company a for now . because i do n't want to impugn anyone . i actually do n't think lehman was any worse or better than any of the other players here . so when they go into bankruptcy , something very interesting happens . now , bank b , they were already worried about these cdos . these cdos were already an issue . and they were probably thinking , boy when when loan c comes due , i 'm going to be in trouble . or when loan d , or f , or whatever , i 'm going to be in trouble because i 'm going to be in that situation that i 'm essentially forcing bank a into right now . but now i have a new problem . this loan to bank a is n't getting paid off . and who knows ? bank a is going to go into bankruptcy . maybe in bankruptcy we realize that these are worth nothing . and if those are worth nothing , then maybe i 'm very junior in seniority in terms of where my loan is and maybe i get nothing . or i get a few pennies on the dollar here . so maybe i thought this was $ 1 billion and i have to write this down to $ 0.5 billion . so now i have two problems . i have this and i have this . and once again , this is a non liquid loan . bank a is in bankruptcy . and if i wanted to somehow get the value of this i have to wait for all of bank a 's assets to go into liquidation . and then whatever assets i get i would have to sell it . so this is kind of a frozen asset . so once again , i 'm stuck holding this non liquid asset . so now i have this non liquid asset that 's probably not worth what i thought it was , which was a loan to a . then i also have these cdos . and now , god forbid , let 's say that i had another loan to bank d. and now let 's say bank d goes bankrupt . and then i have another loan that 's bad on top of these cdos . but the cdos were the crux of the issue . that 's what caused the situation . if bank a could have only sold this cdo for $ 2 billion , it would n't have caused this chain reaction . and lehman brothers really was the thing that catalyzed this whole chain of events . and then you can imagine now bank c is worried because now bank b has all of these illiquid assets on top of these cdos and it starts to look bad . and you can imagine , now it 's even less likely that when a bank , let 's say that bank d is the next one to go into a dire situation , it 's even less likely that bank d can get a loan from a third bank . because all the banks are getting scared now . all the banks are saying , i 'm not going to loan money to anyone . if i can get any cash from anybody i 'm just going to keep it . so that when it 's my turn , when the market starts looking at me , i at least have a little cash . so everyone is frozen . everyone wants to collect their loans from everyone else and no one wants to give loans to anybody else . so that 's the situation we 're in . and that 's the difficulty that the fed is somehow trying to unwind . and i realized i 'm out of time again . i will confront that issue in the next video .
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and they probably are doing just fine with the bonuses they 've collected after sourcing these cdos for the past eight years or five years or however long . but what i want to show you in this video is what people are talking about when they say systemic risk . so these $ 4 billion in liabilities , these are loans , maybe from other banks .
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so what is the systemic risk ?
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i think we 're now ready to tackle the big picture and what has our government officials so worried right now . so what i 've done is , i 've just drawn the balance sheets for a bunch of banks . obviously , this is simplified . and i made all of their balance sheets look the same . all of these banks , each of these kind of represents the balance sheet of a bank . and just to explain it , the left-hand side of this balance sheet , so this column right here -- and maybe i can , at least for the first bank , mark it a little bit . so what i 'm squaring off in magenta , that 's the assets of that bank . what i 'm squaring off in blue , that 's the liabilities of the bank . and what i wrote here is , it has $ 4 billion of liabilities . its assets , i divided it between $ 3 billion of other assets and $ 2 billion of cdos . because we want to focus on the cdos , because that 's the crux of everything that 's going on . and we have $ 5 billion in assets , $ 4 billion of liabilities , so you have $ 1 billion in equity . so that 's what 's left there . so this is just another visual representation that liabilities plus equity is equal to assets . or assets minus liabilities is equal to equity . and i 've just copied and pasted this one balance sheet a bunch of times . i do n't know whether we 're going to use all those . but let 's just assume , for simplicity , that a ton of banks in the system have this identical balance sheet . obviously , they do n't have an identical balance sheet . but all of their balance sheets might have kind of similar properties . this is n't always the case , different banks have different exposures to cdos . some of them have a lot , some of them have a little bit . some of them are valuing them more conservatively than others . but just for the sake of simplicity , i 've just made all the banks in the situation where the book value of the cdos that they have on their balance sheets is the larger than their equity value . and i did that for a reason . because it leads to the issue of , are these banks facing just a liquidity issue or are they facing just a solvency issue ? if you believe that these are worth $ 3 billion , these assets , these liabilities are worth $ 4 billion , then the crux of whether it 's a liquidity or a solvency issue all falls down as to whether these are worth $ 2 billion or not . for example , if these are worth $ 2 billion , then you have $ 1 billion of equity . if these are worth $ 1.5 billion , well maybe they 're being a little optimistic here , but you 'll still have $ 0.5 billion of equity . so you 're still solvent . and in that situation , in theory , one is just if they do n't have the cash when some of their debt comes due , they should just be able to borrow some money and get past that hurdle . and then in the future maybe sell their assets and still have positive equity . however , if the true value of those cdos , and this is kind of a philosophical question , what 's the true value of anything ? and the best thing that we as humans have been to be able to come up with is a market . the market value tends to be the best representation of the true value of something . let 's say the true value of this is $ 1 billion or less , then we have a situation . for example , if these are worth nothing , then we only have $ 3 billion of assets , $ 4 billion of liabilities , we have negative equity . this company is worth nothing . and to lend this bank or this company any money would just be throwing good money after bad . because that money is just going to go into a black hole . because one of the people who this company owes money to is probably not going to see their money . and if you are the most junior person lending the money -- which means that when all the money is distributed if they go into bankruptcy , you 're the last person to see the money -- then you 're just throwing good money after bad . so that 's the issue . but i want you to see the big picture now . because if it was just an issue with one bank it would n't be a big deal . if it was just bear stearns or if it was just lehman brothers , not a big deal , let the greedy bankers go bankrupt . and they probably are doing just fine with the bonuses they 've collected after sourcing these cdos for the past eight years or five years or however long . but what i want to show you in this video is what people are talking about when they say systemic risk . so these $ 4 billion in liabilities , these are loans , maybe from other banks . in fact , probably from other banks . and those loans from other banks , those are assets of other banks . for example , let 's say this is bank a , this is bank b . maybe a billion of these are a loan from bank b . and if this is a loan from bank b , bank b would have an asset called loan to bank a . on bank b 's balance sheet we 're calling this a loan to bank a . this is one of its assets . and then one of its liabilities will be a loan from bank b . so how can i say this ? they took this money and they gave it to b. i 'm sorry , b had money , gave it to a in the form of a loan . and so that cash ended up here . and they got an asset called loan to bank a . and this is a liability , loan from bank b . and they might have taken that money and they might have lent it to bank c down here . i think you 're starting to see how this gets pretty hairy very fast . so let 's say that bank a , one of its $ 3 billion in assets , is a loan to bank c. and so on bank c 's balance sheet , it 'll say loan from bank a . or so we owe a $ 1 billion . and a says , c owes me $ 1 billion , and that 's all fine . and then you see that oh , we owe b $ 1 billion . and then we could keep doing this . or i could just even make this into a circle already . so maybe bank b has some money that it owes to someone else . and let 's say that someone else , just for fun , just to make this interesting -- i think you can extrapolate and think about how this gets complicated very fast . bank b has borrowed money from bank c. so bank c will have an asset here that says , no i lent money to bank b . fair enough . ok , so now we 're in an interesting situation . let 's say this loan , the loan from bank b to bank a comes due . and we 've studied this multiple times . and let 's say for whatever reason , all of these other loans , they 're not liquid . they 're not due yet . so bank a ca n't get rid of these loans . so let 's say this comes due , this is $ 4 billion . they ca n't sell any of this . so bank a has to come up with $ 1 billion somehow for bank b . so that 's the situation we 're dealing with . i 'm just going to say that they ca n't sell any of these assets . so it all comes down to the cdos . so there 's a couple of issues here . if you think it is just an issue of illiquidity , if these are $ 2 billion of assets , they 're really worth $ 2 billion , but bank a just ca n't sell them . because either there 's quote-unquote nobody willing to buy . although , i would argue if no-one is willing to buy something , then its true value is probably zero . but let 's just say bank a says no-one is willing to buy , we 're just illiquid , this is really worth $ 2 billion . so one situation is they could get a loan from someone . maybe the fed would be willing to take this as collateral . so they would give this as collateral to the fed . maybe the fed will give them a billion dollar loan . and then they can use that to pay bank b . let 's say that 's off the table because this is just smelly enough collateral that not even the fed , which we now realize is willing to do anything to support the markets , not even the fed is willing to give them a loan . or enough of a loan to pay off that loan . the other situation is maybe they can get an equity infusion from a sovereign wealth fund . and we covered that a couple of videos ago . where the sovereign wealth fund will essentially inject some cash . it 'll dilute the shares and then you know maybe we had 500 million shares before . now we 'll have 2 billion shares . so the sovereign wealth fund will take over roughly 80 % of the company . and in exchange for 80 % of the company , would give maybe $ 2 billion and then you could use that to pay off this loan . but let 's say that that 's not on the table anymore either . because the sovereign wealth funds have gotten burned so much . so what happens ? well we learned what happens . if you ca n't get a loan , a new loan , to replace this loan , or if you ca n't get an equity infusion from kind of a greater fool , what happens ? you go into bankruptcy . and this is what happened to lehman brothers . lehman brothers went into bankruptcy . no sovereign wealth fund , no one else bought the company . and i should probably do another video on that scenario . and they could n't get a loan . so they went bankrupt . i should call this company l actually . but i 'll call it company a for now . because i do n't want to impugn anyone . i actually do n't think lehman was any worse or better than any of the other players here . so when they go into bankruptcy , something very interesting happens . now , bank b , they were already worried about these cdos . these cdos were already an issue . and they were probably thinking , boy when when loan c comes due , i 'm going to be in trouble . or when loan d , or f , or whatever , i 'm going to be in trouble because i 'm going to be in that situation that i 'm essentially forcing bank a into right now . but now i have a new problem . this loan to bank a is n't getting paid off . and who knows ? bank a is going to go into bankruptcy . maybe in bankruptcy we realize that these are worth nothing . and if those are worth nothing , then maybe i 'm very junior in seniority in terms of where my loan is and maybe i get nothing . or i get a few pennies on the dollar here . so maybe i thought this was $ 1 billion and i have to write this down to $ 0.5 billion . so now i have two problems . i have this and i have this . and once again , this is a non liquid loan . bank a is in bankruptcy . and if i wanted to somehow get the value of this i have to wait for all of bank a 's assets to go into liquidation . and then whatever assets i get i would have to sell it . so this is kind of a frozen asset . so once again , i 'm stuck holding this non liquid asset . so now i have this non liquid asset that 's probably not worth what i thought it was , which was a loan to a . then i also have these cdos . and now , god forbid , let 's say that i had another loan to bank d. and now let 's say bank d goes bankrupt . and then i have another loan that 's bad on top of these cdos . but the cdos were the crux of the issue . that 's what caused the situation . if bank a could have only sold this cdo for $ 2 billion , it would n't have caused this chain reaction . and lehman brothers really was the thing that catalyzed this whole chain of events . and then you can imagine now bank c is worried because now bank b has all of these illiquid assets on top of these cdos and it starts to look bad . and you can imagine , now it 's even less likely that when a bank , let 's say that bank d is the next one to go into a dire situation , it 's even less likely that bank d can get a loan from a third bank . because all the banks are getting scared now . all the banks are saying , i 'm not going to loan money to anyone . if i can get any cash from anybody i 'm just going to keep it . so that when it 's my turn , when the market starts looking at me , i at least have a little cash . so everyone is frozen . everyone wants to collect their loans from everyone else and no one wants to give loans to anybody else . so that 's the situation we 're in . and that 's the difficulty that the fed is somehow trying to unwind . and i realized i 'm out of time again . i will confront that issue in the next video .
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and let 's say that someone else , just for fun , just to make this interesting -- i think you can extrapolate and think about how this gets complicated very fast . bank b has borrowed money from bank c. so bank c will have an asset here that says , no i lent money to bank b . fair enough .
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if the debt is owned to bank b , does n't it mean that bank b and c owes each other , so their debts cancel each other out ?
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i think we 're now ready to tackle the big picture and what has our government officials so worried right now . so what i 've done is , i 've just drawn the balance sheets for a bunch of banks . obviously , this is simplified . and i made all of their balance sheets look the same . all of these banks , each of these kind of represents the balance sheet of a bank . and just to explain it , the left-hand side of this balance sheet , so this column right here -- and maybe i can , at least for the first bank , mark it a little bit . so what i 'm squaring off in magenta , that 's the assets of that bank . what i 'm squaring off in blue , that 's the liabilities of the bank . and what i wrote here is , it has $ 4 billion of liabilities . its assets , i divided it between $ 3 billion of other assets and $ 2 billion of cdos . because we want to focus on the cdos , because that 's the crux of everything that 's going on . and we have $ 5 billion in assets , $ 4 billion of liabilities , so you have $ 1 billion in equity . so that 's what 's left there . so this is just another visual representation that liabilities plus equity is equal to assets . or assets minus liabilities is equal to equity . and i 've just copied and pasted this one balance sheet a bunch of times . i do n't know whether we 're going to use all those . but let 's just assume , for simplicity , that a ton of banks in the system have this identical balance sheet . obviously , they do n't have an identical balance sheet . but all of their balance sheets might have kind of similar properties . this is n't always the case , different banks have different exposures to cdos . some of them have a lot , some of them have a little bit . some of them are valuing them more conservatively than others . but just for the sake of simplicity , i 've just made all the banks in the situation where the book value of the cdos that they have on their balance sheets is the larger than their equity value . and i did that for a reason . because it leads to the issue of , are these banks facing just a liquidity issue or are they facing just a solvency issue ? if you believe that these are worth $ 3 billion , these assets , these liabilities are worth $ 4 billion , then the crux of whether it 's a liquidity or a solvency issue all falls down as to whether these are worth $ 2 billion or not . for example , if these are worth $ 2 billion , then you have $ 1 billion of equity . if these are worth $ 1.5 billion , well maybe they 're being a little optimistic here , but you 'll still have $ 0.5 billion of equity . so you 're still solvent . and in that situation , in theory , one is just if they do n't have the cash when some of their debt comes due , they should just be able to borrow some money and get past that hurdle . and then in the future maybe sell their assets and still have positive equity . however , if the true value of those cdos , and this is kind of a philosophical question , what 's the true value of anything ? and the best thing that we as humans have been to be able to come up with is a market . the market value tends to be the best representation of the true value of something . let 's say the true value of this is $ 1 billion or less , then we have a situation . for example , if these are worth nothing , then we only have $ 3 billion of assets , $ 4 billion of liabilities , we have negative equity . this company is worth nothing . and to lend this bank or this company any money would just be throwing good money after bad . because that money is just going to go into a black hole . because one of the people who this company owes money to is probably not going to see their money . and if you are the most junior person lending the money -- which means that when all the money is distributed if they go into bankruptcy , you 're the last person to see the money -- then you 're just throwing good money after bad . so that 's the issue . but i want you to see the big picture now . because if it was just an issue with one bank it would n't be a big deal . if it was just bear stearns or if it was just lehman brothers , not a big deal , let the greedy bankers go bankrupt . and they probably are doing just fine with the bonuses they 've collected after sourcing these cdos for the past eight years or five years or however long . but what i want to show you in this video is what people are talking about when they say systemic risk . so these $ 4 billion in liabilities , these are loans , maybe from other banks . in fact , probably from other banks . and those loans from other banks , those are assets of other banks . for example , let 's say this is bank a , this is bank b . maybe a billion of these are a loan from bank b . and if this is a loan from bank b , bank b would have an asset called loan to bank a . on bank b 's balance sheet we 're calling this a loan to bank a . this is one of its assets . and then one of its liabilities will be a loan from bank b . so how can i say this ? they took this money and they gave it to b. i 'm sorry , b had money , gave it to a in the form of a loan . and so that cash ended up here . and they got an asset called loan to bank a . and this is a liability , loan from bank b . and they might have taken that money and they might have lent it to bank c down here . i think you 're starting to see how this gets pretty hairy very fast . so let 's say that bank a , one of its $ 3 billion in assets , is a loan to bank c. and so on bank c 's balance sheet , it 'll say loan from bank a . or so we owe a $ 1 billion . and a says , c owes me $ 1 billion , and that 's all fine . and then you see that oh , we owe b $ 1 billion . and then we could keep doing this . or i could just even make this into a circle already . so maybe bank b has some money that it owes to someone else . and let 's say that someone else , just for fun , just to make this interesting -- i think you can extrapolate and think about how this gets complicated very fast . bank b has borrowed money from bank c. so bank c will have an asset here that says , no i lent money to bank b . fair enough . ok , so now we 're in an interesting situation . let 's say this loan , the loan from bank b to bank a comes due . and we 've studied this multiple times . and let 's say for whatever reason , all of these other loans , they 're not liquid . they 're not due yet . so bank a ca n't get rid of these loans . so let 's say this comes due , this is $ 4 billion . they ca n't sell any of this . so bank a has to come up with $ 1 billion somehow for bank b . so that 's the situation we 're dealing with . i 'm just going to say that they ca n't sell any of these assets . so it all comes down to the cdos . so there 's a couple of issues here . if you think it is just an issue of illiquidity , if these are $ 2 billion of assets , they 're really worth $ 2 billion , but bank a just ca n't sell them . because either there 's quote-unquote nobody willing to buy . although , i would argue if no-one is willing to buy something , then its true value is probably zero . but let 's just say bank a says no-one is willing to buy , we 're just illiquid , this is really worth $ 2 billion . so one situation is they could get a loan from someone . maybe the fed would be willing to take this as collateral . so they would give this as collateral to the fed . maybe the fed will give them a billion dollar loan . and then they can use that to pay bank b . let 's say that 's off the table because this is just smelly enough collateral that not even the fed , which we now realize is willing to do anything to support the markets , not even the fed is willing to give them a loan . or enough of a loan to pay off that loan . the other situation is maybe they can get an equity infusion from a sovereign wealth fund . and we covered that a couple of videos ago . where the sovereign wealth fund will essentially inject some cash . it 'll dilute the shares and then you know maybe we had 500 million shares before . now we 'll have 2 billion shares . so the sovereign wealth fund will take over roughly 80 % of the company . and in exchange for 80 % of the company , would give maybe $ 2 billion and then you could use that to pay off this loan . but let 's say that that 's not on the table anymore either . because the sovereign wealth funds have gotten burned so much . so what happens ? well we learned what happens . if you ca n't get a loan , a new loan , to replace this loan , or if you ca n't get an equity infusion from kind of a greater fool , what happens ? you go into bankruptcy . and this is what happened to lehman brothers . lehman brothers went into bankruptcy . no sovereign wealth fund , no one else bought the company . and i should probably do another video on that scenario . and they could n't get a loan . so they went bankrupt . i should call this company l actually . but i 'll call it company a for now . because i do n't want to impugn anyone . i actually do n't think lehman was any worse or better than any of the other players here . so when they go into bankruptcy , something very interesting happens . now , bank b , they were already worried about these cdos . these cdos were already an issue . and they were probably thinking , boy when when loan c comes due , i 'm going to be in trouble . or when loan d , or f , or whatever , i 'm going to be in trouble because i 'm going to be in that situation that i 'm essentially forcing bank a into right now . but now i have a new problem . this loan to bank a is n't getting paid off . and who knows ? bank a is going to go into bankruptcy . maybe in bankruptcy we realize that these are worth nothing . and if those are worth nothing , then maybe i 'm very junior in seniority in terms of where my loan is and maybe i get nothing . or i get a few pennies on the dollar here . so maybe i thought this was $ 1 billion and i have to write this down to $ 0.5 billion . so now i have two problems . i have this and i have this . and once again , this is a non liquid loan . bank a is in bankruptcy . and if i wanted to somehow get the value of this i have to wait for all of bank a 's assets to go into liquidation . and then whatever assets i get i would have to sell it . so this is kind of a frozen asset . so once again , i 'm stuck holding this non liquid asset . so now i have this non liquid asset that 's probably not worth what i thought it was , which was a loan to a . then i also have these cdos . and now , god forbid , let 's say that i had another loan to bank d. and now let 's say bank d goes bankrupt . and then i have another loan that 's bad on top of these cdos . but the cdos were the crux of the issue . that 's what caused the situation . if bank a could have only sold this cdo for $ 2 billion , it would n't have caused this chain reaction . and lehman brothers really was the thing that catalyzed this whole chain of events . and then you can imagine now bank c is worried because now bank b has all of these illiquid assets on top of these cdos and it starts to look bad . and you can imagine , now it 's even less likely that when a bank , let 's say that bank d is the next one to go into a dire situation , it 's even less likely that bank d can get a loan from a third bank . because all the banks are getting scared now . all the banks are saying , i 'm not going to loan money to anyone . if i can get any cash from anybody i 'm just going to keep it . so that when it 's my turn , when the market starts looking at me , i at least have a little cash . so everyone is frozen . everyone wants to collect their loans from everyone else and no one wants to give loans to anybody else . so that 's the situation we 're in . and that 's the difficulty that the fed is somehow trying to unwind . and i realized i 'm out of time again . i will confront that issue in the next video .
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and let 's say that someone else , just for fun , just to make this interesting -- i think you can extrapolate and think about how this gets complicated very fast . bank b has borrowed money from bank c. so bank c will have an asset here that says , no i lent money to bank b . fair enough .
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bank b loaning to a , and a loaning to c , then c loaning to b ... is this called floating ?
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( piano playing ) dr. steven zucker : we 're looking at one of the great sandro botticelli 's and also one of the most enigmatic , the primavera . dr. beth harris : which means spring . in the center we see venus in her sacred grove looking directly out at us . dr. zucker : the figures in the foreground are parted to allow venus an unobstructed view of us and for us to look back at her and perhaps even to enter into the space . dr. harris : the trees around her part to show us the sky , so there 's almost a sense of a halo around her . dr. zucker : it 's true , there 's a half circle . actually , i read that as almost architectural , almost as an apps and it reminds us that usually what we would find in a space like this from the renaissance would be the virgin mary in an ecclesiastic environment , but here we have a natural or mythic environment and we have venus . dr. harris : right . i mean , here we are . we 're in the renaissance . one definition of the renaissance is that it 's a rebirth of ancient greek and roman culture and here we have an artist who 's embracing a pagan subject , the subject of venus . and also other elements from ancient greek and roman mythology . yeah . dr. zucker : lots of ancient greek and roman figures . dr. harris : we have the three graces on the left . dr. zucker : so , let 's talk about who they are for just a sec . this is a subject that was very popular in roman statuary and it was an opportunity that allowed for a sculptor to show the human body from three sides simultaneously , so that is you multiply the figure and you just turn them slightly each time so that you really see a figure in the round . dr. harris : and then on the far left , we have the god mars , who 's the god of war . he 's put away his weapon . dr. zucker : he 's at peace in her garden . dr. harris : who would n't be at peace in her garden ? look at it . it 's fabulous and we 're not sure exactly what he 's doing . he 's got a stick in his hand . he may be pushing away the clouds that appear to be coming in from the left . dr. zucker : only a sunny day in paradise . dr. harris : absolutely . and then on the right , we have three more figures , zephyr , a god of the wind , who is ... dr. zucker : he is ... that 's the blue figure . dr. harris : that 's the blue figure who is abducting the figure of chloris who , you can see , has a branch with leaves coming out of her mouth that collides with the figure next to her who is the figure of flora . so , they may be one in the same person . dr. zucker : in other words , the actual abduction of chloris might actually result in flora and what flora is doing here , is she 's reaching into her satchel , which is full of blossoms , which she seems to be strewing or sewing on this , sort of , carpet of foliage below . this is , after all , primavera . this is spring . dr. harris : spring . dr. zucker : yeah . dr. harris : so , there 's a sense of the fertility of nature . dr. zucker : there 's one other figure , which is venus ' son just above her , blindfolded . this is , of course , cupid , who 's about to unleash his arrow on one of the unwitting graces and , of course , he does n't know who he 's going to hit , but we can sort of figure it out . dr. harris : typical of botticelli , we have figures who are elongated , weightless , who stand in rather impossible positions . things that we do n't normally expect from renaissance art . dr. zucker : so , this really is at odds with many of the traditions that we learn about when it comes to the 15th century . this is not a painting that 's about linear perspective . there 's a little bit of atmospheric perspective that can be seen in the traces of landscape between the trees , but beyond that this is a very frontal painting . it 's very much a freeze and it very much is referencing what we think might be a literary set of ideas . art historians really do n't know what this painting is about and we 've been looking for texts that it might refer to . dr. harris : and , in a way , it does n't really matter to the throngs of people who come to see it and to me because it 's incredibly beautiful and it may be that because it has no specific meaning , it 's easier for us in the 21st century to enjoy it . dr. zucker : there are lots of passages here that are just , i think , glorious . if you look at the daffiness quality of the drape that protect the graces , for instance , and the tassels there . they 're just beautiful . i 'm especially taken where the hands of the graces come together in those three places , creating a kind of wonderful complexity and beauty and just a kind of visual invention that is playful and an expression of a kind of complex notion of beauty . one of the ways in which this painting is understood is it 's possibly as a sort of neo-platonic treatise or a kind of meditation on different kinds of beauty . dr. harris : venus herself is astoundingly beautiful . she tilts her head to one side and holds up her drapery and motions with her hand and looks directly at us . and in a way it 's impossible not to want to join her in the garden . ( piano playing )
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dr. zucker : there 's one other figure , which is venus ' son just above her , blindfolded . this is , of course , cupid , who 's about to unleash his arrow on one of the unwitting graces and , of course , he does n't know who he 's going to hit , but we can sort of figure it out . dr. harris : typical of botticelli , we have figures who are elongated , weightless , who stand in rather impossible positions .
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what is on the tip of cupid 's arrow ?
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( piano playing ) dr. steven zucker : we 're looking at one of the great sandro botticelli 's and also one of the most enigmatic , the primavera . dr. beth harris : which means spring . in the center we see venus in her sacred grove looking directly out at us . dr. zucker : the figures in the foreground are parted to allow venus an unobstructed view of us and for us to look back at her and perhaps even to enter into the space . dr. harris : the trees around her part to show us the sky , so there 's almost a sense of a halo around her . dr. zucker : it 's true , there 's a half circle . actually , i read that as almost architectural , almost as an apps and it reminds us that usually what we would find in a space like this from the renaissance would be the virgin mary in an ecclesiastic environment , but here we have a natural or mythic environment and we have venus . dr. harris : right . i mean , here we are . we 're in the renaissance . one definition of the renaissance is that it 's a rebirth of ancient greek and roman culture and here we have an artist who 's embracing a pagan subject , the subject of venus . and also other elements from ancient greek and roman mythology . yeah . dr. zucker : lots of ancient greek and roman figures . dr. harris : we have the three graces on the left . dr. zucker : so , let 's talk about who they are for just a sec . this is a subject that was very popular in roman statuary and it was an opportunity that allowed for a sculptor to show the human body from three sides simultaneously , so that is you multiply the figure and you just turn them slightly each time so that you really see a figure in the round . dr. harris : and then on the far left , we have the god mars , who 's the god of war . he 's put away his weapon . dr. zucker : he 's at peace in her garden . dr. harris : who would n't be at peace in her garden ? look at it . it 's fabulous and we 're not sure exactly what he 's doing . he 's got a stick in his hand . he may be pushing away the clouds that appear to be coming in from the left . dr. zucker : only a sunny day in paradise . dr. harris : absolutely . and then on the right , we have three more figures , zephyr , a god of the wind , who is ... dr. zucker : he is ... that 's the blue figure . dr. harris : that 's the blue figure who is abducting the figure of chloris who , you can see , has a branch with leaves coming out of her mouth that collides with the figure next to her who is the figure of flora . so , they may be one in the same person . dr. zucker : in other words , the actual abduction of chloris might actually result in flora and what flora is doing here , is she 's reaching into her satchel , which is full of blossoms , which she seems to be strewing or sewing on this , sort of , carpet of foliage below . this is , after all , primavera . this is spring . dr. harris : spring . dr. zucker : yeah . dr. harris : so , there 's a sense of the fertility of nature . dr. zucker : there 's one other figure , which is venus ' son just above her , blindfolded . this is , of course , cupid , who 's about to unleash his arrow on one of the unwitting graces and , of course , he does n't know who he 's going to hit , but we can sort of figure it out . dr. harris : typical of botticelli , we have figures who are elongated , weightless , who stand in rather impossible positions . things that we do n't normally expect from renaissance art . dr. zucker : so , this really is at odds with many of the traditions that we learn about when it comes to the 15th century . this is not a painting that 's about linear perspective . there 's a little bit of atmospheric perspective that can be seen in the traces of landscape between the trees , but beyond that this is a very frontal painting . it 's very much a freeze and it very much is referencing what we think might be a literary set of ideas . art historians really do n't know what this painting is about and we 've been looking for texts that it might refer to . dr. harris : and , in a way , it does n't really matter to the throngs of people who come to see it and to me because it 's incredibly beautiful and it may be that because it has no specific meaning , it 's easier for us in the 21st century to enjoy it . dr. zucker : there are lots of passages here that are just , i think , glorious . if you look at the daffiness quality of the drape that protect the graces , for instance , and the tassels there . they 're just beautiful . i 'm especially taken where the hands of the graces come together in those three places , creating a kind of wonderful complexity and beauty and just a kind of visual invention that is playful and an expression of a kind of complex notion of beauty . one of the ways in which this painting is understood is it 's possibly as a sort of neo-platonic treatise or a kind of meditation on different kinds of beauty . dr. harris : venus herself is astoundingly beautiful . she tilts her head to one side and holds up her drapery and motions with her hand and looks directly at us . and in a way it 's impossible not to want to join her in the garden . ( piano playing )
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it 's fabulous and we 're not sure exactly what he 's doing . he 's got a stick in his hand . he may be pushing away the clouds that appear to be coming in from the left .
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is this the artist the guy in the movie the pick up artist is referencing whey he uses the line `` you 've got the face of a botticelli ?
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( piano playing ) dr. steven zucker : we 're looking at one of the great sandro botticelli 's and also one of the most enigmatic , the primavera . dr. beth harris : which means spring . in the center we see venus in her sacred grove looking directly out at us . dr. zucker : the figures in the foreground are parted to allow venus an unobstructed view of us and for us to look back at her and perhaps even to enter into the space . dr. harris : the trees around her part to show us the sky , so there 's almost a sense of a halo around her . dr. zucker : it 's true , there 's a half circle . actually , i read that as almost architectural , almost as an apps and it reminds us that usually what we would find in a space like this from the renaissance would be the virgin mary in an ecclesiastic environment , but here we have a natural or mythic environment and we have venus . dr. harris : right . i mean , here we are . we 're in the renaissance . one definition of the renaissance is that it 's a rebirth of ancient greek and roman culture and here we have an artist who 's embracing a pagan subject , the subject of venus . and also other elements from ancient greek and roman mythology . yeah . dr. zucker : lots of ancient greek and roman figures . dr. harris : we have the three graces on the left . dr. zucker : so , let 's talk about who they are for just a sec . this is a subject that was very popular in roman statuary and it was an opportunity that allowed for a sculptor to show the human body from three sides simultaneously , so that is you multiply the figure and you just turn them slightly each time so that you really see a figure in the round . dr. harris : and then on the far left , we have the god mars , who 's the god of war . he 's put away his weapon . dr. zucker : he 's at peace in her garden . dr. harris : who would n't be at peace in her garden ? look at it . it 's fabulous and we 're not sure exactly what he 's doing . he 's got a stick in his hand . he may be pushing away the clouds that appear to be coming in from the left . dr. zucker : only a sunny day in paradise . dr. harris : absolutely . and then on the right , we have three more figures , zephyr , a god of the wind , who is ... dr. zucker : he is ... that 's the blue figure . dr. harris : that 's the blue figure who is abducting the figure of chloris who , you can see , has a branch with leaves coming out of her mouth that collides with the figure next to her who is the figure of flora . so , they may be one in the same person . dr. zucker : in other words , the actual abduction of chloris might actually result in flora and what flora is doing here , is she 's reaching into her satchel , which is full of blossoms , which she seems to be strewing or sewing on this , sort of , carpet of foliage below . this is , after all , primavera . this is spring . dr. harris : spring . dr. zucker : yeah . dr. harris : so , there 's a sense of the fertility of nature . dr. zucker : there 's one other figure , which is venus ' son just above her , blindfolded . this is , of course , cupid , who 's about to unleash his arrow on one of the unwitting graces and , of course , he does n't know who he 's going to hit , but we can sort of figure it out . dr. harris : typical of botticelli , we have figures who are elongated , weightless , who stand in rather impossible positions . things that we do n't normally expect from renaissance art . dr. zucker : so , this really is at odds with many of the traditions that we learn about when it comes to the 15th century . this is not a painting that 's about linear perspective . there 's a little bit of atmospheric perspective that can be seen in the traces of landscape between the trees , but beyond that this is a very frontal painting . it 's very much a freeze and it very much is referencing what we think might be a literary set of ideas . art historians really do n't know what this painting is about and we 've been looking for texts that it might refer to . dr. harris : and , in a way , it does n't really matter to the throngs of people who come to see it and to me because it 's incredibly beautiful and it may be that because it has no specific meaning , it 's easier for us in the 21st century to enjoy it . dr. zucker : there are lots of passages here that are just , i think , glorious . if you look at the daffiness quality of the drape that protect the graces , for instance , and the tassels there . they 're just beautiful . i 'm especially taken where the hands of the graces come together in those three places , creating a kind of wonderful complexity and beauty and just a kind of visual invention that is playful and an expression of a kind of complex notion of beauty . one of the ways in which this painting is understood is it 's possibly as a sort of neo-platonic treatise or a kind of meditation on different kinds of beauty . dr. harris : venus herself is astoundingly beautiful . she tilts her head to one side and holds up her drapery and motions with her hand and looks directly at us . and in a way it 's impossible not to want to join her in the garden . ( piano playing )
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this is not a painting that 's about linear perspective . there 's a little bit of atmospheric perspective that can be seen in the traces of landscape between the trees , but beyond that this is a very frontal painting . it 's very much a freeze and it very much is referencing what we think might be a literary set of ideas .
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does the audio seem a bit funky ?
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( lively music ) dr. zucker : we 're in the pompidou in paris and we 're looking at l�szl� moholy-nagy . this is a '20 from 1924 . moholy-nagy was a member of the hungarian avant-garde but in 1920 , he comes to dessau , to germany to walter gropius ' bauhaus and takes over the first year program . now , what 's really important is that when moholy-nagy comes in , he comes in almost as a kind of engineer . he 's often portrayed in work coveralls and he helps the bauhaus transform into a school that emphasizes the industrial to a much greater extent . dr. harris : with the arrival of moholy-nagy , we have this new interest in the machine and we certainly have a sense here of very simplified forms . it 's easy to misunderstand its simplicity , unless one spend some time with it . dr. zucker : okay , so at first , it simply looks like a number of geometric forms that are overlapping and there 's nothing much there . but in fact , this is a really careful study about space transparency , translucency and opacity . dr. harris : so , if you think about it in terms of light , it becomes easier to understand its complexity . dr. zucker : and this is in fact , one of the so called light paintings . let 's see if we can work our way through it . my eye is led into this canvas by this long plane of glass or what seems like glass . this purely transparent form that almost looks like an outsize glass slide that you might use under a microscope . dr. harris : and it forms a diagonal line that suggests a recession into space . dr. zucker : i wan na stay with that metaphor of the microscope 's glass slide for a moment because i think that there is a kind of scientific investigation here . dr. harris : so , if we have that transparent , glass-like shape that forms that diagonal , we have another similar form that does n't appear transparent that emerges into our space almost like it 's abutting against the transparent shape . dr. zucker : but not exactly at a right angle , right ? it 's a bit more open and it goes into a much deeper space . dr. harris : and it 's remarkable to me how deep a space , moholy-nagy has constructed just with these very , very simple forms . we also have a sense of opacity and transparency and translucency in the forms around the circle that are overlapping here and also , in the two vertical forms . dr. zucker : well , what 's interesting about those vertical forms , is that instead of using orthogonals to create space , he 's using scale to create space . so , we have the larger , thicker one and then evidently , much deeper in space , much further away , the one that 's more narrow . dr. harris : and also that circle in the distance that helps to create an illusion of space . dr. zucker : right , and then look at the bands both vertical and horizontal that crossed . you know , those are translucent but when they crossed , in a sense there 's enough visual mass so they become opaque but then counter that with what we might take to be opacity but it 's not . it 's reflectivity in a way that the transparent plane actually overlays that translucent vertical and then you have a kind of white negative space dr. harris : ( laughs ) as opposed to opacity . dr. harris : so , we have the opaque , which one ca n't see through . the translucent , which one can see through somewhat . the transparent , which one can see through entirely and reflectivity and the different ways that those overlap and affect color and space . what 's interesting to me is that moholy-nagy has not represented any of those things . if you think about the way that painters represent reflectivity and mirrors or transparency with a wine glass in a still life , all of those things are still here but in a very different language . dr. zucker : well , it 's almost the language of mathematics . this is an abstraction that refers to those things in the purest terms , almost in mathematical terms , as opposed to the representation of those things . ( lively music )
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dr. zucker : well , it 's almost the language of mathematics . this is an abstraction that refers to those things in the purest terms , almost in mathematical terms , as opposed to the representation of those things . ( lively music )
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could art works of this very abstract kind have been influenced by other things outside the art world such as literature ?
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( lively music ) dr. zucker : we 're in the pompidou in paris and we 're looking at l�szl� moholy-nagy . this is a '20 from 1924 . moholy-nagy was a member of the hungarian avant-garde but in 1920 , he comes to dessau , to germany to walter gropius ' bauhaus and takes over the first year program . now , what 's really important is that when moholy-nagy comes in , he comes in almost as a kind of engineer . he 's often portrayed in work coveralls and he helps the bauhaus transform into a school that emphasizes the industrial to a much greater extent . dr. harris : with the arrival of moholy-nagy , we have this new interest in the machine and we certainly have a sense here of very simplified forms . it 's easy to misunderstand its simplicity , unless one spend some time with it . dr. zucker : okay , so at first , it simply looks like a number of geometric forms that are overlapping and there 's nothing much there . but in fact , this is a really careful study about space transparency , translucency and opacity . dr. harris : so , if you think about it in terms of light , it becomes easier to understand its complexity . dr. zucker : and this is in fact , one of the so called light paintings . let 's see if we can work our way through it . my eye is led into this canvas by this long plane of glass or what seems like glass . this purely transparent form that almost looks like an outsize glass slide that you might use under a microscope . dr. harris : and it forms a diagonal line that suggests a recession into space . dr. zucker : i wan na stay with that metaphor of the microscope 's glass slide for a moment because i think that there is a kind of scientific investigation here . dr. harris : so , if we have that transparent , glass-like shape that forms that diagonal , we have another similar form that does n't appear transparent that emerges into our space almost like it 's abutting against the transparent shape . dr. zucker : but not exactly at a right angle , right ? it 's a bit more open and it goes into a much deeper space . dr. harris : and it 's remarkable to me how deep a space , moholy-nagy has constructed just with these very , very simple forms . we also have a sense of opacity and transparency and translucency in the forms around the circle that are overlapping here and also , in the two vertical forms . dr. zucker : well , what 's interesting about those vertical forms , is that instead of using orthogonals to create space , he 's using scale to create space . so , we have the larger , thicker one and then evidently , much deeper in space , much further away , the one that 's more narrow . dr. harris : and also that circle in the distance that helps to create an illusion of space . dr. zucker : right , and then look at the bands both vertical and horizontal that crossed . you know , those are translucent but when they crossed , in a sense there 's enough visual mass so they become opaque but then counter that with what we might take to be opacity but it 's not . it 's reflectivity in a way that the transparent plane actually overlays that translucent vertical and then you have a kind of white negative space dr. harris : ( laughs ) as opposed to opacity . dr. harris : so , we have the opaque , which one ca n't see through . the translucent , which one can see through somewhat . the transparent , which one can see through entirely and reflectivity and the different ways that those overlap and affect color and space . what 's interesting to me is that moholy-nagy has not represented any of those things . if you think about the way that painters represent reflectivity and mirrors or transparency with a wine glass in a still life , all of those things are still here but in a very different language . dr. zucker : well , it 's almost the language of mathematics . this is an abstraction that refers to those things in the purest terms , almost in mathematical terms , as opposed to the representation of those things . ( lively music )
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( lively music ) dr. zucker : we 're in the pompidou in paris and we 're looking at l�szl� moholy-nagy . this is a '20 from 1924 .
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what is the sculpture/painting on the left ?
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we 're told carbon-14 is an element which loses exactly half of its mass every 5730 years . the mass of a sample of carbon-14 can be modeled by a function , m , which depends on its age , t , in years . we measure that the initial mass of a sample of carbon-14 is 741 grams . write a function that models the mass of the carbon-14 sample remaining t years since the initial measurement . alright , so , like always , pause the video and see if you can come up with this function , m , that is going to be a function of t , the years since the initial measurement . alright , let 's work through it together . what i like to do is , i always like to start off with a little bit of a table to get a sense of things . so let 's think about t , how much time , how many years have passed since the initial measurement , and what the amount of mass we 're going to have . well , we know that the initial , we know that the initial mass of a sample of carbon-14 is 741 grams , so at t equals zero , our mass is 741 . now , what 's another interesting t that we could think about ? well , we know at every 5730 years , we lose exactly half of our mass of carbon-14 . every 5730 years . so let 's think about what happens when t is 5730 . well , we 're going to lose half of our mass , so we 're going to multiply this times 1/2 . so this is going to be 741 times 1/2 . i 'm not even gon na calculate what that is right now . and then let 's say we have another 5730 years take place , so that 's going to be , and i 'm just gon na write two times 5730 . i could calculate what it 's going to be . 10,000 , 11,460 or something like that . alright , but let 's just go with two times 5730 . is it 10 ? yeah . 10,000 plus 1400 so 11,400 plus 60 . yeah . so 11,460 . but let 's just leave it like this . well , then , it 's gon na be this times 1/2 . so it 's gon na be 741 times 1/2 times 1/2 . so we 're gon na multiply by 1/2 again . and so this is the same thing as 741 times 1/2 squared . and then , let 's just think about if we wait another 5730 years , so three times 5730 . well , then it 's going to be 1/2 times this . so it 's going to be 741 , this times 1/2 is gon na be 1/2 to the third power . so you might notice a little bit of a pattern here . however many half-lifes we have , we 're gon na multiply , we 're gon na raise 1/2 to that power and then we multiply it times our initial mass . this is one half-life has gone by , two half-lifes , we have an exponent of two , three half-lifes , we multiply by three . sorry , we multiply by 1/2 three times . so what 's going to be a general way to express m of t ? well , m of t is going to be our initial value , 741 , times , and you might already be identifying this as an exponential function , we 're going to multiply times this number , which we could call our common ratio , as many half-lifes has passed by . so how do we know how many half-lives have passed by ? well , we could take t , and we could divide it by the half-life . and try to test this out . when t equals zero , it 's gon na be 1/2 to the zeroth power , which is just one , and we 're just gon na have 741 . when t is equal to 5730 , this exponent is going to be one , which we want it to be . we 're just gon na multiply our initial value by 1/2 once . when this exponent is two times 5730 , when t is two times 5730 , well then the exponent is going to be two , and we 're gon na multiply by 1/2 twice . it 's going to be 1/2 to the second power . and it 's going to work for everything in between . when we are a fraction of a half-life along , we 're gon na get a non-integer exponent , and that , too , will work out . and so this is our function . we are , we are done . we have written our function , m , that models the mass of carbon-14 remaining t years since the initial measurement .
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we 're told carbon-14 is an element which loses exactly half of its mass every 5730 years . the mass of a sample of carbon-14 can be modeled by a function , m , which depends on its age , t , in years .
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if carbon-14 's mass gets halved every 5730 years , then would n't carbon-14 never disappear ?
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we 're told carbon-14 is an element which loses exactly half of its mass every 5730 years . the mass of a sample of carbon-14 can be modeled by a function , m , which depends on its age , t , in years . we measure that the initial mass of a sample of carbon-14 is 741 grams . write a function that models the mass of the carbon-14 sample remaining t years since the initial measurement . alright , so , like always , pause the video and see if you can come up with this function , m , that is going to be a function of t , the years since the initial measurement . alright , let 's work through it together . what i like to do is , i always like to start off with a little bit of a table to get a sense of things . so let 's think about t , how much time , how many years have passed since the initial measurement , and what the amount of mass we 're going to have . well , we know that the initial , we know that the initial mass of a sample of carbon-14 is 741 grams , so at t equals zero , our mass is 741 . now , what 's another interesting t that we could think about ? well , we know at every 5730 years , we lose exactly half of our mass of carbon-14 . every 5730 years . so let 's think about what happens when t is 5730 . well , we 're going to lose half of our mass , so we 're going to multiply this times 1/2 . so this is going to be 741 times 1/2 . i 'm not even gon na calculate what that is right now . and then let 's say we have another 5730 years take place , so that 's going to be , and i 'm just gon na write two times 5730 . i could calculate what it 's going to be . 10,000 , 11,460 or something like that . alright , but let 's just go with two times 5730 . is it 10 ? yeah . 10,000 plus 1400 so 11,400 plus 60 . yeah . so 11,460 . but let 's just leave it like this . well , then , it 's gon na be this times 1/2 . so it 's gon na be 741 times 1/2 times 1/2 . so we 're gon na multiply by 1/2 again . and so this is the same thing as 741 times 1/2 squared . and then , let 's just think about if we wait another 5730 years , so three times 5730 . well , then it 's going to be 1/2 times this . so it 's going to be 741 , this times 1/2 is gon na be 1/2 to the third power . so you might notice a little bit of a pattern here . however many half-lifes we have , we 're gon na multiply , we 're gon na raise 1/2 to that power and then we multiply it times our initial mass . this is one half-life has gone by , two half-lifes , we have an exponent of two , three half-lifes , we multiply by three . sorry , we multiply by 1/2 three times . so what 's going to be a general way to express m of t ? well , m of t is going to be our initial value , 741 , times , and you might already be identifying this as an exponential function , we 're going to multiply times this number , which we could call our common ratio , as many half-lifes has passed by . so how do we know how many half-lives have passed by ? well , we could take t , and we could divide it by the half-life . and try to test this out . when t equals zero , it 's gon na be 1/2 to the zeroth power , which is just one , and we 're just gon na have 741 . when t is equal to 5730 , this exponent is going to be one , which we want it to be . we 're just gon na multiply our initial value by 1/2 once . when this exponent is two times 5730 , when t is two times 5730 , well then the exponent is going to be two , and we 're gon na multiply by 1/2 twice . it 's going to be 1/2 to the second power . and it 's going to work for everything in between . when we are a fraction of a half-life along , we 're gon na get a non-integer exponent , and that , too , will work out . and so this is our function . we are , we are done . we have written our function , m , that models the mass of carbon-14 remaining t years since the initial measurement .
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when t is equal to 5730 , this exponent is going to be one , which we want it to be . we 're just gon na multiply our initial value by 1/2 once . when this exponent is two times 5730 , when t is two times 5730 , well then the exponent is going to be two , and we 're gon na multiply by 1/2 twice .
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what is the constant value for mercury 194 ?
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we 're told carbon-14 is an element which loses exactly half of its mass every 5730 years . the mass of a sample of carbon-14 can be modeled by a function , m , which depends on its age , t , in years . we measure that the initial mass of a sample of carbon-14 is 741 grams . write a function that models the mass of the carbon-14 sample remaining t years since the initial measurement . alright , so , like always , pause the video and see if you can come up with this function , m , that is going to be a function of t , the years since the initial measurement . alright , let 's work through it together . what i like to do is , i always like to start off with a little bit of a table to get a sense of things . so let 's think about t , how much time , how many years have passed since the initial measurement , and what the amount of mass we 're going to have . well , we know that the initial , we know that the initial mass of a sample of carbon-14 is 741 grams , so at t equals zero , our mass is 741 . now , what 's another interesting t that we could think about ? well , we know at every 5730 years , we lose exactly half of our mass of carbon-14 . every 5730 years . so let 's think about what happens when t is 5730 . well , we 're going to lose half of our mass , so we 're going to multiply this times 1/2 . so this is going to be 741 times 1/2 . i 'm not even gon na calculate what that is right now . and then let 's say we have another 5730 years take place , so that 's going to be , and i 'm just gon na write two times 5730 . i could calculate what it 's going to be . 10,000 , 11,460 or something like that . alright , but let 's just go with two times 5730 . is it 10 ? yeah . 10,000 plus 1400 so 11,400 plus 60 . yeah . so 11,460 . but let 's just leave it like this . well , then , it 's gon na be this times 1/2 . so it 's gon na be 741 times 1/2 times 1/2 . so we 're gon na multiply by 1/2 again . and so this is the same thing as 741 times 1/2 squared . and then , let 's just think about if we wait another 5730 years , so three times 5730 . well , then it 's going to be 1/2 times this . so it 's going to be 741 , this times 1/2 is gon na be 1/2 to the third power . so you might notice a little bit of a pattern here . however many half-lifes we have , we 're gon na multiply , we 're gon na raise 1/2 to that power and then we multiply it times our initial mass . this is one half-life has gone by , two half-lifes , we have an exponent of two , three half-lifes , we multiply by three . sorry , we multiply by 1/2 three times . so what 's going to be a general way to express m of t ? well , m of t is going to be our initial value , 741 , times , and you might already be identifying this as an exponential function , we 're going to multiply times this number , which we could call our common ratio , as many half-lifes has passed by . so how do we know how many half-lives have passed by ? well , we could take t , and we could divide it by the half-life . and try to test this out . when t equals zero , it 's gon na be 1/2 to the zeroth power , which is just one , and we 're just gon na have 741 . when t is equal to 5730 , this exponent is going to be one , which we want it to be . we 're just gon na multiply our initial value by 1/2 once . when this exponent is two times 5730 , when t is two times 5730 , well then the exponent is going to be two , and we 're gon na multiply by 1/2 twice . it 's going to be 1/2 to the second power . and it 's going to work for everything in between . when we are a fraction of a half-life along , we 're gon na get a non-integer exponent , and that , too , will work out . and so this is our function . we are , we are done . we have written our function , m , that models the mass of carbon-14 remaining t years since the initial measurement .
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well , we know at every 5730 years , we lose exactly half of our mass of carbon-14 . every 5730 years . so let 's think about what happens when t is 5730 .
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what equation would give us the amount of mercury remaining after t years ?
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we 're told carbon-14 is an element which loses exactly half of its mass every 5730 years . the mass of a sample of carbon-14 can be modeled by a function , m , which depends on its age , t , in years . we measure that the initial mass of a sample of carbon-14 is 741 grams . write a function that models the mass of the carbon-14 sample remaining t years since the initial measurement . alright , so , like always , pause the video and see if you can come up with this function , m , that is going to be a function of t , the years since the initial measurement . alright , let 's work through it together . what i like to do is , i always like to start off with a little bit of a table to get a sense of things . so let 's think about t , how much time , how many years have passed since the initial measurement , and what the amount of mass we 're going to have . well , we know that the initial , we know that the initial mass of a sample of carbon-14 is 741 grams , so at t equals zero , our mass is 741 . now , what 's another interesting t that we could think about ? well , we know at every 5730 years , we lose exactly half of our mass of carbon-14 . every 5730 years . so let 's think about what happens when t is 5730 . well , we 're going to lose half of our mass , so we 're going to multiply this times 1/2 . so this is going to be 741 times 1/2 . i 'm not even gon na calculate what that is right now . and then let 's say we have another 5730 years take place , so that 's going to be , and i 'm just gon na write two times 5730 . i could calculate what it 's going to be . 10,000 , 11,460 or something like that . alright , but let 's just go with two times 5730 . is it 10 ? yeah . 10,000 plus 1400 so 11,400 plus 60 . yeah . so 11,460 . but let 's just leave it like this . well , then , it 's gon na be this times 1/2 . so it 's gon na be 741 times 1/2 times 1/2 . so we 're gon na multiply by 1/2 again . and so this is the same thing as 741 times 1/2 squared . and then , let 's just think about if we wait another 5730 years , so three times 5730 . well , then it 's going to be 1/2 times this . so it 's going to be 741 , this times 1/2 is gon na be 1/2 to the third power . so you might notice a little bit of a pattern here . however many half-lifes we have , we 're gon na multiply , we 're gon na raise 1/2 to that power and then we multiply it times our initial mass . this is one half-life has gone by , two half-lifes , we have an exponent of two , three half-lifes , we multiply by three . sorry , we multiply by 1/2 three times . so what 's going to be a general way to express m of t ? well , m of t is going to be our initial value , 741 , times , and you might already be identifying this as an exponential function , we 're going to multiply times this number , which we could call our common ratio , as many half-lifes has passed by . so how do we know how many half-lives have passed by ? well , we could take t , and we could divide it by the half-life . and try to test this out . when t equals zero , it 's gon na be 1/2 to the zeroth power , which is just one , and we 're just gon na have 741 . when t is equal to 5730 , this exponent is going to be one , which we want it to be . we 're just gon na multiply our initial value by 1/2 once . when this exponent is two times 5730 , when t is two times 5730 , well then the exponent is going to be two , and we 're gon na multiply by 1/2 twice . it 's going to be 1/2 to the second power . and it 's going to work for everything in between . when we are a fraction of a half-life along , we 're gon na get a non-integer exponent , and that , too , will work out . and so this is our function . we are , we are done . we have written our function , m , that models the mass of carbon-14 remaining t years since the initial measurement .
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well , then , it 's gon na be this times 1/2 . so it 's gon na be 741 times 1/2 times 1/2 . so we 're gon na multiply by 1/2 again .
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why did n't you multiply 741 by 1.5 instead of 0.5 ?
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the 4 in the number 5,634 is blank times blank than the 4 in the number 12,749 . so let 's think about what they 're saying . so the 4 in the number 5,634 , that 's literally in the ones place . it literally just represents 4 . now , the 4 in the number 12,749 , that 4 is in the tens place . it represents 40 . so this 4 , it 's a 10 times smaller value than this 4 . or this 4 , i should say . this 4 right over here represents 4 , while this represents 40 . so it is 10 times smaller than the 4 in 12,749 . 4 by itself is 10 times smaller than 40 . make sure i got the right answer . let 's do another one . in the number 3,779,264 , how many times less is the value of the second 7 than the value of the first 7 ? how many times less is the value of the second 7 than the value of the first 7 ? so the second 7 right over here , that 's in the ten thousands place . it literally represents 70,000 , 7 ten thousands , or 70,000 , while this represents 700 thousands , or 700,000 . so the second 7 is 1/10 the value of the first 7 . or another way of thinking , it 's 10 times less . this is 70,000 , and this is 700,000 . so the value of the second 7 is 10 times less than the value of the first 7 . let 's do one more . this is fun . fill in the following blanks to complete the relationships between 25,430 and 2,543 . all right , so 25,430 is 10 times larger than 2,543 . literally , you take this , you multiply it by 10 , you 're going to get 25,430 . the digits in 25,430 are one place to the blank of the digits in the number 2,543 . well , let 's think about it . here you have a 2 in the thousands place . here you have a 2 in the ten thousands place . here you have a 5 in the hundreds place . here you have a 5 in the thousands place . and we could keep going . but what we see is a corresponding digit . in 25,430 , they 're one place to the left of the digits right over here . now , finally , we 're going to take 25,430 and divide it by 2,543 . well , we already know that the first number is 10 times larger than this number right over here . so literally , if you divide the smaller number into the larger one , you 're going to get 10 . this right over here is 10 times larger than this . so that was the first part of the question , and we are done .
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here you have a 5 in the hundreds place . here you have a 5 in the thousands place . and we could keep going . but what we see is a corresponding digit .
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what else could you tell the viewers about understanding place value ?
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the 4 in the number 5,634 is blank times blank than the 4 in the number 12,749 . so let 's think about what they 're saying . so the 4 in the number 5,634 , that 's literally in the ones place . it literally just represents 4 . now , the 4 in the number 12,749 , that 4 is in the tens place . it represents 40 . so this 4 , it 's a 10 times smaller value than this 4 . or this 4 , i should say . this 4 right over here represents 4 , while this represents 40 . so it is 10 times smaller than the 4 in 12,749 . 4 by itself is 10 times smaller than 40 . make sure i got the right answer . let 's do another one . in the number 3,779,264 , how many times less is the value of the second 7 than the value of the first 7 ? how many times less is the value of the second 7 than the value of the first 7 ? so the second 7 right over here , that 's in the ten thousands place . it literally represents 70,000 , 7 ten thousands , or 70,000 , while this represents 700 thousands , or 700,000 . so the second 7 is 1/10 the value of the first 7 . or another way of thinking , it 's 10 times less . this is 70,000 , and this is 700,000 . so the value of the second 7 is 10 times less than the value of the first 7 . let 's do one more . this is fun . fill in the following blanks to complete the relationships between 25,430 and 2,543 . all right , so 25,430 is 10 times larger than 2,543 . literally , you take this , you multiply it by 10 , you 're going to get 25,430 . the digits in 25,430 are one place to the blank of the digits in the number 2,543 . well , let 's think about it . here you have a 2 in the thousands place . here you have a 2 in the ten thousands place . here you have a 5 in the hundreds place . here you have a 5 in the thousands place . and we could keep going . but what we see is a corresponding digit . in 25,430 , they 're one place to the left of the digits right over here . now , finally , we 're going to take 25,430 and divide it by 2,543 . well , we already know that the first number is 10 times larger than this number right over here . so literally , if you divide the smaller number into the larger one , you 're going to get 10 . this right over here is 10 times larger than this . so that was the first part of the question , and we are done .
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the 4 in the number 5,634 is blank times blank than the 4 in the number 12,749 . so let 's think about what they 're saying .
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so the 25430 can be more than 2543 although they 're almost the same numbers ?
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the 4 in the number 5,634 is blank times blank than the 4 in the number 12,749 . so let 's think about what they 're saying . so the 4 in the number 5,634 , that 's literally in the ones place . it literally just represents 4 . now , the 4 in the number 12,749 , that 4 is in the tens place . it represents 40 . so this 4 , it 's a 10 times smaller value than this 4 . or this 4 , i should say . this 4 right over here represents 4 , while this represents 40 . so it is 10 times smaller than the 4 in 12,749 . 4 by itself is 10 times smaller than 40 . make sure i got the right answer . let 's do another one . in the number 3,779,264 , how many times less is the value of the second 7 than the value of the first 7 ? how many times less is the value of the second 7 than the value of the first 7 ? so the second 7 right over here , that 's in the ten thousands place . it literally represents 70,000 , 7 ten thousands , or 70,000 , while this represents 700 thousands , or 700,000 . so the second 7 is 1/10 the value of the first 7 . or another way of thinking , it 's 10 times less . this is 70,000 , and this is 700,000 . so the value of the second 7 is 10 times less than the value of the first 7 . let 's do one more . this is fun . fill in the following blanks to complete the relationships between 25,430 and 2,543 . all right , so 25,430 is 10 times larger than 2,543 . literally , you take this , you multiply it by 10 , you 're going to get 25,430 . the digits in 25,430 are one place to the blank of the digits in the number 2,543 . well , let 's think about it . here you have a 2 in the thousands place . here you have a 2 in the ten thousands place . here you have a 5 in the hundreds place . here you have a 5 in the thousands place . and we could keep going . but what we see is a corresponding digit . in 25,430 , they 're one place to the left of the digits right over here . now , finally , we 're going to take 25,430 and divide it by 2,543 . well , we already know that the first number is 10 times larger than this number right over here . so literally , if you divide the smaller number into the larger one , you 're going to get 10 . this right over here is 10 times larger than this . so that was the first part of the question , and we are done .
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in the number 3,779,264 , how many times less is the value of the second 7 than the value of the first 7 ? how many times less is the value of the second 7 than the value of the first 7 ? so the second 7 right over here , that 's in the ten thousands place .
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what is the largest place value ?
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the 4 in the number 5,634 is blank times blank than the 4 in the number 12,749 . so let 's think about what they 're saying . so the 4 in the number 5,634 , that 's literally in the ones place . it literally just represents 4 . now , the 4 in the number 12,749 , that 4 is in the tens place . it represents 40 . so this 4 , it 's a 10 times smaller value than this 4 . or this 4 , i should say . this 4 right over here represents 4 , while this represents 40 . so it is 10 times smaller than the 4 in 12,749 . 4 by itself is 10 times smaller than 40 . make sure i got the right answer . let 's do another one . in the number 3,779,264 , how many times less is the value of the second 7 than the value of the first 7 ? how many times less is the value of the second 7 than the value of the first 7 ? so the second 7 right over here , that 's in the ten thousands place . it literally represents 70,000 , 7 ten thousands , or 70,000 , while this represents 700 thousands , or 700,000 . so the second 7 is 1/10 the value of the first 7 . or another way of thinking , it 's 10 times less . this is 70,000 , and this is 700,000 . so the value of the second 7 is 10 times less than the value of the first 7 . let 's do one more . this is fun . fill in the following blanks to complete the relationships between 25,430 and 2,543 . all right , so 25,430 is 10 times larger than 2,543 . literally , you take this , you multiply it by 10 , you 're going to get 25,430 . the digits in 25,430 are one place to the blank of the digits in the number 2,543 . well , let 's think about it . here you have a 2 in the thousands place . here you have a 2 in the ten thousands place . here you have a 5 in the hundreds place . here you have a 5 in the thousands place . and we could keep going . but what we see is a corresponding digit . in 25,430 , they 're one place to the left of the digits right over here . now , finally , we 're going to take 25,430 and divide it by 2,543 . well , we already know that the first number is 10 times larger than this number right over here . so literally , if you divide the smaller number into the larger one , you 're going to get 10 . this right over here is 10 times larger than this . so that was the first part of the question , and we are done .
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the 4 in the number 5,634 is blank times blank than the 4 in the number 12,749 . so let 's think about what they 're saying .
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is n't `` and '' only used in decimals and fractions ?
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the 4 in the number 5,634 is blank times blank than the 4 in the number 12,749 . so let 's think about what they 're saying . so the 4 in the number 5,634 , that 's literally in the ones place . it literally just represents 4 . now , the 4 in the number 12,749 , that 4 is in the tens place . it represents 40 . so this 4 , it 's a 10 times smaller value than this 4 . or this 4 , i should say . this 4 right over here represents 4 , while this represents 40 . so it is 10 times smaller than the 4 in 12,749 . 4 by itself is 10 times smaller than 40 . make sure i got the right answer . let 's do another one . in the number 3,779,264 , how many times less is the value of the second 7 than the value of the first 7 ? how many times less is the value of the second 7 than the value of the first 7 ? so the second 7 right over here , that 's in the ten thousands place . it literally represents 70,000 , 7 ten thousands , or 70,000 , while this represents 700 thousands , or 700,000 . so the second 7 is 1/10 the value of the first 7 . or another way of thinking , it 's 10 times less . this is 70,000 , and this is 700,000 . so the value of the second 7 is 10 times less than the value of the first 7 . let 's do one more . this is fun . fill in the following blanks to complete the relationships between 25,430 and 2,543 . all right , so 25,430 is 10 times larger than 2,543 . literally , you take this , you multiply it by 10 , you 're going to get 25,430 . the digits in 25,430 are one place to the blank of the digits in the number 2,543 . well , let 's think about it . here you have a 2 in the thousands place . here you have a 2 in the ten thousands place . here you have a 5 in the hundreds place . here you have a 5 in the thousands place . and we could keep going . but what we see is a corresponding digit . in 25,430 , they 're one place to the left of the digits right over here . now , finally , we 're going to take 25,430 and divide it by 2,543 . well , we already know that the first number is 10 times larger than this number right over here . so literally , if you divide the smaller number into the larger one , you 're going to get 10 . this right over here is 10 times larger than this . so that was the first part of the question , and we are done .
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fill in the following blanks to complete the relationships between 25,430 and 2,543 . all right , so 25,430 is 10 times larger than 2,543 . literally , you take this , you multiply it by 10 , you 're going to get 25,430 .
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how is 25,430 10 times lager than 2543 when 2543 is not in the 10 thousands place ?
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