Datasets:
jacobcd52
/
hs_math_cleaned

Dataset card Data Studio Files
xet
 Community
1
text
stringlengths
235
3.08k
1.7a + 3b = 11.1 1.2c + 0.3d = 3.9 and –0.2c – 0.4d = 0.4 ax + cy = 6 and 3ax – 2cy = –42 ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up RRRRRound Up ound Up ound Up The elimination method is usually more reliable than trying to solve complicated systems of equations graphically. Just be really careful w...
.3 tions tions Equa Equa Systems of Systems of tions Equations Systems of Equa tions Equa Systems of Systems of tions tions Equa Equa Systems of Systems of Equations Systems of Equa tions Equa Systems of tions Systems of tions tions pplica pplica β€” β€” β€” β€” β€” AAAAApplica tions pplications tions pplica tions tions pplica p...
+ d = 15 β€” this is y st equationtiontiontiontion st equa st equa our fir our fir β€” this is y β€” this is y our first equa β€” this is your fir st equa our fir β€” this is y Now, consider the fact that each CD cost $8.50, each DVD cost $12.50, and Akemi spent $163.50. This leads to the equation: 8.50c + 12.50d = 163.50 β€” thi...
Practice 1. Pedro bought a total of 18 paperback and hardcover books for a total of $150. If each paperback was on sale for $6.50 and the hardcovers were on sale for $9.50 each, calculate how many paperbacks and how many hardcovers Pedro bought. 2. Three cans of tuna fish and four cans of corned beef cost $12.50. Howe...
system of equations is: x + y = 40 y – 4x = –15 Now solve the system of equations. The y-coefficients are the same, so subtract one equation from the other to eliminate y: y + x = 40 – y – 4x = –15 5x = 55 fi x = 11 Now substitute 11 for x in an original equation. x + y = 40 11 + y = 40 y = 29 Therefore Jose is 11 year...
) inches 14 4. By finding the values of x and y, calculate the area of the rectangle on the left. All dimensions are in inches. (3x + 5) inches 5. Given that the triangle on the right is an isosceles triangle with a 40 cm perimeter, find the values of x and y 2x + 3y ound Up ound Up RRRRRound Up ound Up ound Up ound Up...
Systems of tions Equations Systems of Equa tions Equa Systems of Systems of tions tions Equa Equa Systems of Systems of Equations tions Systems of Equa tions Equa Systems of Systems of lems lems er Proboboboboblems er Pr er Pr β€” Inteβ€” Inteβ€” Inteβ€” Inteβ€” Integgggger Pr lems lems er Pr er Proboboboboblems β€” Inteβ€” Inteβ€” I...
οΏ½ 38 = 38 β€” True A useful check is to make sure that the answer matches the information given in the question: The sum of the two integers is 15 + 38 = 53. This matches the question. Three times the smaller integer is 15 Γ— 3 = 45. The larger integer is 45 – 38 = 7 less than this. Section 5.3 Section 5.3 Section 5.3 β€” A...
8. In a two-digit number, the sum of the digits is 10. Find the number if it is 36 more than the number formed by reversing its digits. ound Up ound Up RRRRRound Up ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up For questions like this, it doesn’t matter which letters you choose to represent the unknown ...
ixtur What it means for you: You’ll solve percent mix problems involving systems of linear equations. Key words: percent mix system of linear equations Don’t forget: 5 5% is 100, so the number of gallons of real fruit juice in a gallons of apple drink is 0.05a. tions tions Equa Equa Systems of Systems of tions Equation...
a + 5s = 100 Now solve the system of equations. The coefficient of a is the same in both equations, so subtract one equation from the other to eliminate a. a + 5s = 100 – a + s = 50 4s = 50 fi s = 12.5 Substitute 12.5 for s in one of the original equations. a + s = 50 a + 12.5 = 50 a = 37.5 That means that 37.5 gallons...
must be mixed to produce 138 pounds of a mixture that would be worth 80 cents per pound? 6. A pharmacist has a bottle of 10% boric acid and a bottle of 6% boric acid. A prescription requires 50 milliliters of a 7% boric acid solution. What volume of each solution should the pharmacist mix to get the desired solution? ...
2% rate of return per year. If the total return from the investments after one year was $200, how much money was invested at each rate? 8. Stephen collected a total of 75 quarters and half-dollars. If he collected $30.50, how many coins of each type did he collect? 9. Robert used his 10% discount card to buy 4 pizzas ...
0: Students a 15.0: Students a 15.0: algalgalgalgalgeeeeebrbrbrbrbraic tec hniques tototototo hniques hniques aic tec aic tec aic techniques hniques aic tec solvsolvsolvsolvsolveeeee rrrrraaaaate pr lems lems te proboboboboblems te pr te pr lems, work lems te pr problems, and percent mixture problems..... What it means...
speed Downriver: When traveling downriver, the water speed adds to the boat speed. So boat speed downriver = x + y Use the formula Distance = Speed Γ— Time: 120 miles = (x + y) Γ— 3 hours 120 = 3(x + y) 120 3 = x + y 40 = x + y Upriver: When traveling upriver, the water speed acts against the boat speed. So boat speed u...
way) would take with no wind. 4. A red car and a blue car start at the same time from towns that are 16 miles apart, and travel towards each other. The red car is 7 mph faster than the blue car. After 15 minutes the cars are 7 miles apart. Find the speed of each car. Section 5.3 Section 5.3 Section 5.3 β€” Applications ...
of 25 mph. Marshall leaves the same house 15 minutes later, and drives the same route, but twice as fast as Casey. At what time will Marshall pass Casey, and how far will they be from home when he does? 9. Pittsburgh is 470 miles from Chicago and 350 miles from Philadelphia. Trains leave Chicago and Philadelphia at th...
eggs for 20 cents each. 1) How many days will it be before you break even? Assume you manage to sell all the eggs laid without any going to waste. 2) On one set of axes, draw graphs to show how the costs and amount earned will change over the first 50 days. The β€œbreak-even” point is when the amount of money you have e...
. number of milkshakes you will have earned an amount equal equal ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up RRRRRound Up ound Up ound Up If you come across any real-life situation that involves two or more equations, you’ll probably have to solve them using systems of equations β€” using all the skills...
more variables. For example, 13, 2xΒ², and –x3yn4 are all monomials. A Polynomial Can Have More Than One Term A polynomial is an algebraic expression that has one or more terms (each of which is a monomial). For example, x + 1 and –3xΒ² + 2x + 1 are polynomials. There are a couple of special types of polynomial: A binom...
omial, binomial, or trinomial. 21. 3x2 + 4 – 8 + x2 23. 3x2y – 2x2y + 8 25. 4x3 + 7 – x3 + 4 – 3x3 – 11 27. 3xy + 4xy + 5x2y – 4xy2 22. 8x3 + x4 – 6x3 + 4 24. 7 – 2y + 3 – 10 26. 5x2 + 9x2 + 4 + 2y 28. 9x5 + 2x2 + 4x4 + 5x5 – 3x4 – x2 Finding the Degree of a Polynomial The degree of a polynomial in x is the size of the...
a2 + 16 3. 42xy 2. 2c – 4a + 6 4. 16a2b + 4ab2 Simplify each of the following polynomials. 5. 0.7x2 + 9.8 – x2 6. 17x2 – 14x9 + 7x9 – 7x2 + 7x9 7. 0.8x4 + 0.3x2 + 9.6 – x2 – 9x4 + 1.6x2 State the degree of the following polynomials. 8. x – 9x6 + 4 9. 14x8 + 16x10 + 4x8 10. 2x2 – 4x4 + 7x5 11. 2x2 – 4x + 8 Simplify each...
. Key words: polynomial like terms inverse Adding Polynomials Adding Polynomials Adding polynomials isn’t difficult at all. The only problem is that you can only add certain parts of each polynomial together. The Opposite of a Polynomial The opposite of a number is its additive inverse. The opposite of a positive numbe...
x – 1 + 3 – 7 = xΒ² + 7x – 5 Method B β€” Vertical Lining Up of Terms –5xΒ² + 3x – 1 + 6xΒ² – x + 3 + 5x – 7 xΒ² + 7x – 5 Both methods give the same solution. Don’t forget: See Topic 1.2.4 for the definition of multiplication. Multiplying a Polynomial by a Number Multiplying a polynomial by a number is the same as adding the...
3 + x – 4) + (x3 – 8) + (4x3 – 3x – 1) 3. (–x6 + x – 5) + (2x6 – 4x – 6) + (–2x6 + 2x – 4) 4. (3x2 – 2x + 7) + (4x2 + 6x – 8) + (–5x2 + 4x – 5) 5. (0.4x3 – 1.1) + (0.3x3 + x – 1.0) + (1.1x3 + 2.1x – 2.0) 6. 7. – 4a3 – 2a + 3 8a4 – 2a3 – 4a + 8 7a4 – 4a – 7 1.1c2 + 1.4c – 0.48 –4.9c2 – 3.6c + 0.98 7.3c2 + 0.13 Multiply ...
just need to combine like terms, then carry out all the subtractions to simplify the expression. Subtracting Polynomials Subtracting polynomials is the same as subtracting numbers. To subtract Polynomial A from Polynomial B, you need to subtract each term of Polynomial A from Polynomial B. Then you can combine any lik...
) = –7xΒ² + x + 5 + 5xΒ² – 3x + 8 = –7xΒ² + 5xΒ² + x – 3x + 5 + 8 = –2xΒ² – 2x + 13 Alternatively, you can do subtraction by lining up terms vertically: Example 3 Subtract –5xΒ² + 3x – 8 from –7xΒ² + x + 5. OR –7xΒ² + x + 5 + (5xΒ² – 3x + 8) –2xΒ² – 2x + 13 This is the opposite of –5xΒ² + 3x – 8 Solution –7xΒ² + x + 5 – (–5xΒ² + 3x...
x – 3x – Independent Practice Subtract the polynomials and simplify the resulting expression. 1. (5a + 8) – (3a + 2) 2. (8x – 2y) – (8x + 4y) 3. (–4x2 + 7x – 3) – (2x2 – 4x + 6) 4. (3a2 + 2a + 6) – (2a2 + a + 3) 5. –3x4 – 2x3 + 4x – 1 – (–2x4 – x3 + 3x2 – 5x + 3) 6. 5 – [(2k + 3) – (3k + 1)] 7. – 10a2 + 4a – 1 – (7a2 +...
Standards: 2.0: Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. 10.0: Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, ...
+ 4x) = 10xΒ² – 6x – 10 + 4xΒ² + 4x = 10xΒ² + 4xΒ² – 6x + 4x – 10 = 14xΒ² – 2x – 10 b) Perimeter of P1 – Perimeter of P2 = (10xΒ² – 6x – 10) – (4xΒ² + 4x) = 10xΒ² – 6x – 10 – 4xΒ² – 4x = 10xΒ² – 4xΒ² – 6x – 4x – 10 = 6xΒ² – 10x – 10 272 Section 6.1 β€” Adding and Subtracting Polynomials Guided Practice Find the perimeter of each of...
5x3 – 5). 10. Find the sum of the opposites of: –2x3 + 3x2 – 5x + 1 and 3x3 – 2x2 + 3x – 3 11. Find the difference between the opposites of: –2x2 – 3x + 5 and 3x2 + 2x – 4 12. Simplify –2(3x2p – 2xp + 1) + 4(2x2p – xp – 3) – 5(x2p – 2xp – 1). 13. If the perimeter of the figure shown is 90 inches, what are the dimensio...
–b (if x Ο€ 0) 4) (cx)b = cbxb a b 5) x = b a x or ( b a ) x 6) x–a = 1 x a (if x Ο€ 0) 7) x0 = 1 Example 1 Simplify the expression (–2x2m)(–3x3m3). Solution (–2x2m)(–3x3m3) = (–2)(–3)(x2)(x3)(m)(m3) = 6x2+3Β·m1+3 = 6x5m4 Put all like variables together Use Rule 1 and add the powers Example 2 Simplify the expression (3a2x...
– b2) 4. 4m2x2(x2 + x + 1) 5. a(a + 4) + 4(a + 4) 6. 2a(a – 4) – 3(a – 4) 7. m2n3(mx2 + 3nx + 2) – 4m2n3 8. 4m2n2(m3n8 + 4) – 3m3n10(m2 + 2n3. b a 6 19 b a Find the value of? that makes these statements true. 10. 11. 10 17 2 7 12. 4 4 20 3 a m 7 a m 6 13. m?(m4 + 2m3) = m6 + 2m5 14. m4a6(3m?a8 + 4m2a?) = 3m7a14 + 4m6a...
using the distributive property. Example 1 Simplify the expression –2a(a + 3a2). Solution –2a(a + 3a2) is a product of the monomial –2a and the binomial (a + 3a2), so multiply each term of the binomial by the monomial. = –2a(a) + (–2a)(3a2) = –2a2 – 6a3 To find the product of two polynomials, such as (a – 2b)(3a + b),...
+ b)(a – b) = a2 – b2 9. (a – b)2 = a2 – 2ab + b2 8. (a + b)2 = a2 + b2 10. (a + b)(a + b) = a2 + 2ab + b2 You Can Multiply Polynomials with Lots of Terms Check it out: Multiply each term in one set of parentheses by every term in the second set of parentheses. Example 5 Simplify (x + 2)(x2 + 2x + 3). Solution (x + 2)...
(a2 – b2)(a + ab + b) = a3 – b3 You Can Also Use the Stacking Method You can find the product of 63 and 27 by β€œstacking” the two numbers and doing long multiplication: Don’t forget: The units, tens, hundreds, and thousands are in separate columns Γ— 63 2 Γ— 63 You can use the same idea to find the products of polynomial...
. (2x + 7)(3x + 5) 6. (2x – 4y)(3x – 3y + 4) Use the stack method to multiply. Show all your work. 7. (x2 – 4)(x + 3) 9. (4x2 – 5x)(1 + 2x – 3x2) 8. (x – y)(3x2 + xy + y2) 10. (x + 4)(3x2 – 2x + 5) Use these formulas to find each of the products in Exercises 11–16. (a + b)2 = a2 + 2ab + b2 (a – b)2 = a2 – 2ab + b2 (a +...
of the middle rectangle is 5x + 6 – 2x = (3x + 6) inches. The width of the middle rectangle is 3x + 2 – 2x = (x + 2) inches. x Area of space = area of large rectangle – area of small rectangle = (5x + 6)(3x + 2) – (3x + 6)(x + 2) = 15x2 + 10x + 18x + 12 – (3x2 + 6x + 6x + 12) = 15x2 + 28x + 12 – 3x2 – 12x – 12 = (12x2...
x(30x2 + 28x – 32) + 6(30x2 + 28x – 32) Multiply out again = 120x3 + 112x2 – 128x + 180x2 + 168x – 192 = 120x3 + 112x2 + 180x2 – 128x + 168x – 192 Commutative law = (120x3 + 292x2 + 40x – 192) in3 Example 3 Find the volume of a box made from the sheet below by removing the four corners and folding. 2x 2x 2x 2x 2x 2x 2x...
(2x2 + 3x – 1) cm and a height of (3x – 1) cm. 4. A gardener wants to put a walkway around her garden, as shown on the right. What is the area of the walkway? x x x x x (2 + 5) feet (7 + 3) feet x 5. Obike made a box from a 10 inch by 9 inch piece of cardboard by cutting squares of x units from each of the four corner...
Topic 6.2.1 really are useful. In this Topic you’ll use them to divide polynomials by monomials. y a Monomial y a Monomial ynomial b viding a Polololololynomial b ynomial b viding a P DiDiDiDiDividing a P viding a P y a Monomial ynomial by a Monomial y a Monomial ynomial b viding a P To divide a polynomial by a monomi...
utiutiutivvvvve pr the distrib the distrib oper the distrib the distrib = (2x3–1 β—Š y1–1) + (x1–1 β—Š y2–1) = (2x2 β—Š 1) + (1 β—Š y1) = 2x2 + y Section 6.3 Section 6.3 Section 6.3 β€” Dividing Polynomials Section 6.3 Section 6.3 283283283283283 Example Example Example Example Example 33333 Simplify mc v 2 10 4 3 mc v. Solution...
14a7b5 + 10a3b7 by 2a3b4. 7. Divide 4m5x7v6 – 12m4c2x8v4 + 16a3m6c2x9v7 by –4m4x7v4. Find the missing exponent in the quotients. βˆ’ 2 x y? 9. 8 xy 4x ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up RRRRRound Up ound Up ound Up This leads on to the next few Topics, where you’ll divide one polynomial by anot...
solve multistep problems, including word problems, by using these techniques. What it means for you: You’ll learn how to find the multiplicative inverse of a polynomial, and how to use negative exponents. Key words: polynomial monomial reciprocal exponent ynomials and ynomials and PPPPPolololololynomials and ynomials ...
Exponents e Exponents e Exponents ocals as Ne ocals as Ne ecipr ecipr rite R ou Can WWWWWrite R rite R ou Can YYYYYou Can ou Can e Exponents eciprocals as Ne rite Recipr e Exponents ocals as Ne ecipr rite R ou Can This rule of exponents gives you another way to express fractions: x–a = 1 x a provided x Ο€ 0 Example Exa...
out If a pair of parentheses is raised to a negative power, its entire contents go to the bottom of the fraction. Example Example Example Example Example 33333 Simplify the following expressions: a) (a + b)β—Š(b – c)–1 b) (m – c)(m + c)β—Š(m – c)–1 c) (x – 1)–1β—Š(x + 1)–1 Solution Solution Solution Solution Solution a) (a ...
2) –1 15. (z – 5)2(3)–2(15)(z – 5)–1 8. (–2x)2(6ax)–2 10. 4ab Γ— (3a)–1 Γ— (4b)–1 12. (a2b–2)3(a–1b0)4(a–2b–3)–2 14. (3z)–1(z – 2)–2(z2)3(z – 2) 16. 20(z – 3)–2(z + 3)(z – 3)3(6)–1 ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up RRRRRound Up ound Up ound Up Now you’ve got all the tools you need. You’ll start...
oring β€” Fβ€” Fβ€” Fβ€” Fβ€” Factoring actoring actoring actoring actoring Now you’re ready to divide a polynomial by another polynomial. The simplest way to do this is by factoring the numerator and the denominator. essions essions actions Helps to Simplify Expr Canceling Frrrrractions Helps to Simplify Expr actions Helps to S...
+ 3)(x + 4) by ( x + 3 ) 8. 10. Divide 3 x x + by actorededededed actor actor ynomial Can Be F the Polololololynomial Can Be F ynomial Can Be F the P vides Evenlenlenlenlenlyyyyy,,,,, the P the P vides Ev vides Ev It Di IfIfIfIfIf It Di It Di ynomial Can Be Factor It Divides Ev actor ynomial Can Be F the P vides Ev It...
out: See Sections 6.6 and 6.8 for more on factoring quadratics. Guided Practice Simplify the quotients by canceling factors. 11. + x 4 + x 8 2 2 2x 14. x + 12 x 2 x 17. 2 + βˆ’ x +( 2 x 20 ) 5 12. 2x 12 x + 4 16 15. x 9 xβˆ’ 12 2 3 x 18 13. 20 2 z + + 4 5 z z 16. x 5 2 x βˆ’ βˆ’ 15 9 19 20. Find the ratio of the surface area ...
add, subtract, 10.0: Students 10.0: dividevidevidevidevide di di multiply, and di di ynomials..... ynomials ynomials monomials and pol monomials and pol monomials and polynomials monomials and pol ynomials monomials and pol Students solve multistep problems, including word problems, by using these techniques. What it ...
above the line. Leading term of dividend Leading term of divisor 3x 6 Β² – 11 – 11 x x 2 – 5x Then subtract the product of 3x and (2x – 5) from the dividend. 3x groups of (2x – 5) is (6xΒ² – 15x). Subtract 3 x groups of (2 – 5) x from the dividend, leaving a remainder of 4 – 11. x 2 – 5x 3x 6 Β² – 11 – 11 x x x – (6 Β² – ...
omials of degree higher than 2 (such as cubic equations), it’s not always clear how to factor them. You can use long division to help factor expressions. For example, if you divide Polynomial A by Polynomial B and get an answer with remainder zero, then Polynomial B is a factor of Polynomial A. Example Example Example ...
using the long division method. 3 2 x 3. βˆ’ βˆ’ 2 x βˆ’ x 13 3 βˆ’ 6 x 4 2a 4. 2x 2a 7. + 10 + a βˆ’ 1 a 1 βˆ’ 6 8. 2 βˆ’ 2x 3 x x 11 βˆ’ 2 + 9 9. 2 x 15 + βˆ’ x 2 + 5 x 10. 2a 3 1 βˆ’ βˆ’ 4 a + 1 a 11 2a 12. + βˆ’ a 35 βˆ’ a 3 4 36 13. 10 2x + 2 x + 10 x 21 + 3 14. 3 a βˆ’ βˆ’ 7 22 a a βˆ’ βˆ’ 33 a 7 15. The width of a rectangle is (x – 3) cm and th...
multiste ultiste Students solv Students solv Students solv ultiste Students solv luding worororororddddd luding w luding w inc inc lems,,,,, inc lems lems prprprprproboboboboblems including w luding w inc lems prprprprproboboboboblems lems,,,,, b b b b by using these y using these y using these lems lems y using these...
(2x Solution Solution Solution Solution Solution Start by writing out the formula for volume: + 1) c m h cm (4x + 3) cm volume = length Γ— width Γ— height height = volume Γ— length width h = 40 3 x ( 4 + 2 34 + x )( There are a couple of different ways to tackle a division problem like this: 1) You could use long divisio...
: (4x + 3)(2x + 1) = 8xΒ² + 4x + 6x + 3 = 8xΒ² + 10x + 3 Then use long division to divide the volume by the product of the two factors: 8 Β² + 10 + 3 x x 5 – 2 40 Β³ + 34 40 Β³ + 50 Β² + 15 ) x x x –16 Β² – 20 – 6 – (–16 Β² – 20 – 6) x x x x So the answer is h = (5x – 2) cm. 0 Section 6.3 Section 6.3 Section 6.3 β€” Dividing Pol...
) ft. 4. The volume of a prism is (2x3 + x2 – 3x) m3. If the area of the base is (x2 – x) m2, what is the height of the prism? 5. A rectangular prism has volume (b3 + 9b2 + 26b + 24) m3, width (b + 2) m, and length (b + 4) m. Find its height. 6. The volume of a prism is (144s3 + 108s2 – 4s – 3) m3. If the area of the b...
(a + b)Β² expands to give a perfect square trinomial β€” see Topic 6.8.2. Section 6.4 Special Products of Special Products of Two Binomials Two Binomials This Topic is all about special cases of binomial multiplication. Knowing how to expand these special products will save you time when you’re dealing with binomials lat...
When You Expand (a – b)Β², the ab-Term is Negative (a – b)2 = (a – b)(a – b) = a2 – ab – ba + b2 Using the distributive property = a2 – 2ab + b2 You can also relate this equation to the area of a smaller square: a (a – b)2 a (a – b)2 is the same as the area of the darker square. To find the area of the darker square, y...
The expression is already in the form (a + b)(a – b), so you can convert it to the form a2 – b2: (4m + 3)(4m – 3) = (4m)2 – 32 = 16m2 – 9 Guided Practice Find and simplify each product. 21. (m – v)(m + v) 23. (3y + x)(3y – x) 22. (x + 5)(x – 5) 24. (k – 6t)(k + 6t) 25. (3x – 9y)(3x + 9y) 26. (6x + 6y)(6x – 6y) 27. (x ...
2 – b2 and putting in 5y for a and 4 for b you get: (5y – 4) cm Area = (5y + 4)(5y – 4) = (5y)2 – 42 = (25y2 – 16) cm2 Check it out: Using the equation here is a little shorter than expanding the parentheses. 300 Section 6.4 β€” Special Products of Binomials Independent Practice Find the areas of these shapes. 3. 4. 2a –...
6.5.16.5.1 6.5.1 Section 6.5 Monomials Monomials s of s of actor actor FFFFFactor Monomials s of Monomials actors of Monomials s of actor FFFFFactor Monomials Monomials s of s of actor actor s of Monomials actors of Monomials s of Monomials actor California Standards: 11.0 Students applpplpplpplpply basic y basic y ba...
Monomials s β€” and So Do Monomials actor s Has Havvvvve Fe Fe Fe Fe Factor actor s Has Ha Number Number s β€” and So Do Monomials actors β€” and So Do Monomials Numbers Ha s β€” and So Do Monomials actor Number Number Sometimes a number can be written as the product of two or more smaller numbers. Those smaller numbers are c...
302302302302302 Section 6.5 Section 6.5 Section 6.5 β€” Factors Section 6.5 Section 6.5 Example 1 continued The greatest common factor is the product of all the common factors: GCF = = 6xΒ²yΒ² In other words, 6xΒ²yΒ² is the largest possible divisor of 12xΒ²yΒ², 18xΒ³yΒ², and 30x4y4. Guided Practice Use the method from Example 1...
v – 1)3, 12(v – 1)2 21. 6x2yz, 15xz 22. 21x4y4z4, 42x3y4z5, 14x6y3z2 Section 6.5 Section 6.5 Section 6.5 β€” Factors Section 6.5 Section 6.5 303303303303303 Independent Practice Write down all of the factors of each of these numbers: 1. 25 4. 67 3. 36 6. 70 2. 12 5. 80 Write each of these as a product of prime factors: 7...
. If the length of the walkway is 70 ft longer than the width and its area is 6000 ft2, how many paving stones make up the length and the width of the walkway? ound Up ound Up RRRRRound Up ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up Factoring is the best way of working out which smaller parts make up a...
greatest common factor actorsssss actor actor Simple F Simple F Simple Factor actor Simple F Simple F actorsssss actor actor Simple F Simple F Simple Factor actor Simple F Simple F ynomials ynomials ofofofofof P P P P Polololololynomials ynomials ynomials ofofofofof P P P P Polololololynomials ynomials ynomials ynomia...
= (2 Γ— mΒ² Γ— 3 Γ— m) – (2 Γ— mΒ² Γ— 2) = (6 Γ— m2+1) – (4 Γ— mΒ²) = 6m3 – 4mΒ² Section 6.5 Section 6.5 Section 6.5 β€” Factors Section 6.5 Section 6.5 305305305305305 Guided Practice In each polynomial, find the greatest common factor of the terms. 1. 12y2 – 3y 2. a3 + 3a 3. 14a3 – 28a2 + 56a 4. 16y2 – 24y3 5. 60x3 + 24x2 + 16x ...
ided Practice Factor each polynomial below. 13. x2 – 4x 15. 24x3 – 15x2 + 6x 17. 4a3 – 6a2 + 6a 19. 6b3 – 3b2 + 12b 14. x2 – x 16. 8x3 + 2x2 + 4x 18. 14b2 + 7b – 21 20. a4 + a5 + 5a3 306306306306306 Section 6.5 Section 6.5 Section 6.5 β€” Factors Section 6.5 Section 6.5 actors s s s s TTTTToooooooooo actor actor ynomial ...
you’ll have shown that (a + b) is a factor. Take the (a + b) outside parentheses, as above: + ) ad + a ( b ) + bd + a ( b ) ⎞ ⎟⎟⎟⎟ ⎠ = + a ( ) ( ) b b a + ac + ac + bc + ad + bd βŽ› ⎜⎜⎜ ⎝ βŽ› ⎜⎜⎜ ⎝ βŽ›βŽ› ⎜⎜⎜ ⎝ + + ac ( a bcc ) bc + ( a b + + bd ) b ad ( ⎞ ⎟⎟⎟⎟ ⎠ ⎞ ⎟⎟⎟⎟ ⎠ = (a + b)(c + d) Therefore (a + b) is a factor. Sectio...
)(x2 + 2x + x + x + 2) = (x – 2)(x2 + 4x + 2) Guided Practice Factor and simplify. 21. x(2x + 1) + 3(2x + 1) 22. 3y2(2 – 3x) + y(2 – 3x) + 5(2 – 3x) 23. 2x4(5x – 3) – x2(5x – 3) + (5x – 3) 24. 2a(3a – 1) + 6(3a – 1) 25. (4 – x)x2 + (4 – x)2x + (4 – x)1 26. Show that (x + 3) is a factor of x2 + 2x + 3x + 6. 27. Show tha...
m(y – 5)2 – (y – 5) 12. (x2 – 2x) + (4x – 8) 13. (y2 + 3y) + (3y + 9) 14. (2my – 3mx) + (–4y + 6x) 15. –2m(x + 1) + k(x + 1) 16. x5 + 3x3 + 2x4 18. 3x3 – 6x2 + 9x 20. 2m3n – 6m2n2 + 10mn 17. 8y2x + 4yx2 + 4y2x2 19. 4x5 – 4x3 + 16x2 21. Show that (x + 4) is a factor of x2 – 5x + 4x – 20. 22. Show that (2x + 5) is a fac...
oring Quadr actoring Quadr FFFFFactoring Quadr actoring Quadraaaaaticsticsticsticstics actoring Quadr actoring Quadr actoring Quadr California Standards: Students applpplpplpplpply basic y basic y basic Students a Students a 11.0: 11.0: y basic 11.0: Students a 11.0: y basic Students a 11.0: fffffactoring tec hniques t...
or x2 + 12. Some quadratics can be factored β€” in other words they can be expressed as a product of two linear factors. Suppose x2 + bx + c can be written in the form (x + m)(x + n). Then: x2 + bx + c = (x + m)(x + n) b, c, m, and n are numbers = x(x + n) + m(x + n) = x2 + nx + mx + mn Expand out the parentheses using ...
. Example Example Example Example Example 22222 Factor x2 – x – 6. Solution Solution Solution Solution Solution Find two numbers that multiply to give –6 and add to give –1, the coefficient of x. Because c is negative (–6), one number must be positive and the other negative. x2 – x – 6 = (x – 3)(x + 2) Check whether th...
– 8 = x2 + 2x – 8 This is the same as the original expression, so the factors are correct. Section 6.6 Section 6.6 Section 6.6 β€” Factoring Quadratics Section 6.6 Section 6.6 311311311311311 Guided Practice Factor each expression below. 1. a2 + 7a + 10 3. x2 – 17x + 72 5. b2 + 2b – 24 7. x2 – 15x + 54 9. t2 + 16t + 55 ...
n2 – 5n, find the other factors. 23. If (8n – 3) is a factor of 8n3 – 3n2 – 8n + 3, find the other factors. 24. If (2x + 5) is a factor of 2x3 + 15x2 + 13x – 30, find the other factors. 25. If (a – 1) is a factor of a3 – 6a2 + 9a – 4, find the other factors. 26. If (x – 2) is factor of x3 + 5x2 – 32x + 36, find the oth...
pol hese ee polynomials hese ynomials ee pol lude finding lude finding hniques inc hniques inc tectectectectechniques inc lude finding hniques include finding lude finding hniques inc or all or all actor f actor f a common f a common f actor for all a common factor f or all a common f or all actor f a common f ynomial...
using the method in Topic 6.6.1: = 3(x + 2)(x + 3) But β€” if the expression in parentheses still has a Ο€ 1, then the expression will need to be factored using the second method, shown in Example 2. Guided Practice Factor each expression completely. 1. 3x2 + 15x + 12 3. 2t2 – 22t + 60 5. 4x2 + 32x + 64 7. 5m2 + 20m + 15...
if a number is positive then its two factors will be either both positive or both negative. If a number is negative, then its two factors will have different signs β€” one positive and one negative. These facts give important clues about the signs of c1 and c2. Example Example Example Example Example 22222 Factor 3xΒ² + ...
Check it out: Each pair of coefficients c1 and c2 has TWO possible positions. 314314314314314 Section 6.6 Section 6.6 Section 6.6 β€” Factoring Quadratics Section 6.6 Section 6.6 Check it out: You can consider separately whether the values of c1 and c2 should be positive or negative. Guided Practice Factor each polynomi...
and c2 in the parentheses and find a1c2 + a2c1 and a1c2 – a2c1 like before: (6x 1)(x 6) multiplies to give 36x and x, which add/subtract to give 37x or 35x. (6x 6)(x 1) multiplies to give 6x and 6x, which add/subtract to give 12x or 0x. (6x 2)(x 3) multiplies to give 18x and 2x, which add/subtract to give 20x or 16x. ...
3k2 – 2k – 1 49. 18 + 5x – 2x2 51. 9x2 + 12x + 4 53. 3x2 – 7x – 6 55. 6x2 + 2x – 20 57. 6y2 + y – 12 42. 3y2 – y – 2 44. 3x2 – x – 10 46. 2x2 – 5x + 2 48. 3v2 – 16v + 5 50. 28 + x – 2x2 52. 7a2 – 26a – 8 54. 12x2 + 5x – 2 56. 18x2 + x – 4 58. 9m2 – 3m – 20 tic to WWWWWororororork Fk Fk Fk Fk Faster aster aster tic to ...
both the sums and differences of all the different combinations a1c2 and a2c1, you only need to look at the differences. Guided Practice Factor each expression. 59. 2n2 + n – 3 61. 4a2 + 4a + 1 63. 9y2 + 6y + 1 65. 5x2 – x – 18 67. 6t2 + t – 1 60. 2x2 – 5x – 3 62. 3x2 – 4x + 1 64. 4t2 + t – 3 66. 9x2 – 6x + 1 68. b2 +...
next Section you’ll learn about another type β€” quadratic expressions containing two different variables. Section 6.6 Section 6.6 Section 6.6 β€” Factoring Quadratics Section 6.6 Section 6.6 317317317317317 TTTTTopicopicopicopicopic 6.7.16.7.1 6.7.16.7.1 6.7.1 California Standards: Students applpplpplpplpply basic y basi...
ariaariaariaariaariabbbbbleslesleslesles in in in in in TTTTTwwwwwo o o o o VVVVVariaariaariaariaariabbbbbleslesleslesles in in in in So far, most of the quadratics you’ve factored have had only one variable β€” but the same rules apply if there are two variables. actorededededed actor actor Also Be F Also Be F les Can t...
add to make 7pq (3p + 2q)(p + q) β€” this would give pq-terms of 3pq and 2pq, which add to make 5pq 5pq is what you need, so 3pΒ² + 5pq + 2qΒ² = (3p + 2q)(p + q). 318318318318318 Section 6.7 Section 6.7 Section 6.7 β€” More on Factoring Polynomials Section 6.7 Section 6.7 Guided Practice Factor each polynomial below. 1. x2 ...
m – )(m – ) To fill the gaps you need two terms in p that will multiply together to make 5pΒ², and when multiplied by 2m and m respectively, will add together to make –11mp. Try out some sets of parentheses that multiply to make 5pΒ²: (2m – 5p)(m – p) β€” this would give mp-terms of –2mp and –5mp, which add to make –7mp (2...
xz and –12xz, which add to make –6xz (3x – 2z)(3x + 4z) β€” this would give 12xz and –6xz, which add to make +6xz You can stop here because +6xz is the expression you are trying to get. So 9xΒ² + 6xz – 8zΒ² = (3x – 2z)(3x + 4z). Guided Practice Factor each of the polynomials below. 17. 2x2 – 5xy – 3y2 19. 3x2 + 17xy + 10y2...
polynomials. 9. (x2 + 13x + 8) – (3x2 + 10x – 3) – (x2 + 2x + 6) + (4x2 – 5x – 2) 10. (5x2 + 2x + 4) – (6x2 – 3x + 7) + (4x2 – x – 4) 11. (4x2 – 6xy – 10y2) – (2x2 – 8xy + 2y2) 12. (6x2 + 3xy + 8y2) – (3x2 – 12xy – 10y2) 13. (2t2 – 8tz – 5z2) – (4t2 + 2tz – 15z2) + (5t2 + tz – 40z2) 14. (6x2 – 4xy – 25y2) – (2x2 + 4xy...
xy4 26. 16a2z2c + 16abz2c + 4b2z2c Factor and simplify completely. 27. 12x2(x + 2) + 25xy(x + 2) + 12y2(x + 2) 28. 18a2b2(a – 1) – 33ab3(a – 1) – 30b4(a – 1) ound Up ound Up RRRRRound Up ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up This is a long process, so it’s easy to make mistakes. You should always...
in a pol ms in a pol terterms in a pol terter ynomial, ms in a polynomial, ter ynomial, ms in a pol recognizing the difference of two squares, and recognizing perfect squares of binomials. What it means for you: You’ll learn how to factor third-degree expressions. Key words: polynomial degree factor Don’t forget: See ...
⎟⎟ ⎠ Now look at the factor in the parentheses β€” this is a quadratic expression that it may be possible to factor. In this case it’s possible to factor it, using the method from Section 6.6: xΒ³ + 7xΒ² + 12x = x(x2 + 7x + 12) = x(x + 3)(x + 4) Guided Practice Factor completely these polynomials. 1. 4y3 + 26y2 + 40y 3. 6x...
–2 instead of 2. This is helpful because it means that the coefficient of xΒ² becomes 1 β€” which makes the quadratic expression much easier to factor, as you saw in Section 6.6. –2xΒ³ – 2xΒ² + 4x = –2x βŽ› ⎜⎜⎜⎜ ⎝ 2x(x2 + x – 2) Now factor the quadratic: –2xΒ³ – 2xΒ² + 4x = –2x(x2 + x – 2) = –2x(x – 1)(x + 2) ⎞ ⎟⎟⎟⎟ ⎠ Note tha...
288c4d – 128c3d 30. –50y4z – 130y3z + 60y2z 32. –54y3z2 + 21y2z2 + 3yz2 34. 16a4b2 + 176a3b2 + 484a2b2 36. –12b4f2 + 68b3f 2 – 96b2f 2 Section 6.7 Section 6.7 Section 6.7 β€” More on Factoring Polynomials Section 6.7 Section 6.7 323323323323323 Independent Practice Factor these polynomials completely. 1. –8a3 + 78a2 + 2...
+ 32w ∫ 2w(3w – 4)2 13. 12a3b + 70a?b + 72ab ∫ 2ab(2a + 9)(3a + 4) 14. –12a4b? – 58a3b3 – 70a2b3 ∫ –2a2b3(2a + 5)(3a + 7) 15. –56a4b4 + 12a3b4 + 8a2b4 ∫ –?a2b4(2a – 1)(7a + 2) 16. A cylinder has a base with dimensions that are binomial factors. If the volume of the cylinder is (75px3 + 30px2 + 12px) in.3 and the heigh...
the quadratic part. 324324324324324 Section 6.7 Section 6.7 Section 6.7 β€” More on Factoring Polynomials Section 6.7 Section 6.7 TTTTTopicopicopicopicopic 6.8.16.8.1 6.8.16.8.1 6.8.1 California Standards: Students applpplpplpplpply basic y basic y basic Students a Students a 11.0: 11.0: y basic 11.0: Students a 11.0: y...
Being able to recognize the difference of two squares is really useful β€” it helps you factor quadratic expressions. actor Quadraaaaaticsticsticsticstics actor Quadr actor Quadr es to F es to F o Squar ence of TTTTTwwwwwo Squar o Squar ence of Use Difffffferererererence of ence of Use Dif Use Dif es to Factor Quadr o S...
Example Example Example Example Example 22222 Factor 4x2 – 25b2. Solution Solution Solution Solution Solution 4x2 = (2x)2, so the square root of 4x2 is 2x. 25b2 = (5b)2, so the square root of 25b2 is 5b. Put the values into the difference of two squares equation: m2 – c2 = (m + c)(m – c) 4x2 – 25b2 = (2x + 5b)(2x – 5b...