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TeXit
02/27/2022 13:48:10
**omega**
https://cdn.discordapp.c…365540798467.png
245745365540798467.png
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Manan
02/27/2022 13:48:23
Thanks
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omega
02/27/2022 13:48:59
I understand that $x_2$ and $x_4$ are free variables
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TeXit
02/27/2022 13:49:01
**omega**
https://cdn.discordapp.c…365540798467.png
245745365540798467.png
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omega
02/27/2022 13:49:16
but after that I'm having some trouble
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omega
02/27/2022 13:50:54
@Helpers
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omega
02/27/2022 13:54:11
Hello?
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omega
02/27/2022 13:54:19
anybody do linear algebra here?
1
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Gooaaa
02/27/2022 15:37:20
did you rref it?
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Hector Salamanca
02/27/2022 17:04:16
How do I find a polynomial function with a degree of 3, leading coefficient 2, and zeroes 14, 1/2, and 9/2?
1
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help-10
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198,191,547,970,355,200
sills
02/27/2022 17:05:23
uh
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Remavas
02/27/2022 17:06:26
aw, you gave him the answer :/
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392,341,051,794,391,040
Navix
02/27/2022 17:06:38
rip
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603,753,572,412,293,100
Hector Salamanca
02/27/2022 17:06:38
I didn't see it
1
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sills
02/27/2022 17:06:45
good :devilish:
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sills
02/27/2022 17:06:48
we can help explain it now!
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Hector Salamanca
02/27/2022 17:06:49
Also it's she/her please
1
0
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Remavas
02/27/2022 17:07:08
Do you remember how to represent polynomials in terms of their roots?
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Remavas
02/27/2022 17:07:15
~~"we"~~
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Hector Salamanca
02/27/2022 17:07:17
No
1
0
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Hector Salamanca
02/27/2022 17:07:52
I don't remember any of this, I missed school because I had covid
1
0
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Remavas
02/27/2022 17:08:56
Alright, let's take a polynomial of degree 1. Would you agree that we can represent any linear function as $y = c(x-a)$ where c and a are some real numbers
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TeXit
02/27/2022 17:08:59
**Remavas**
https://cdn.discordapp.c…980977336321.png
373884980977336321.png
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Remavas
02/27/2022 17:10:09
(And do you know polynomial division, because that would make the explanation even easier)
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Hector Salamanca
02/27/2022 17:10:18
I don't know I never learned any of this
1
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Remavas
02/27/2022 17:10:49
Well ok, we can represent any linear function in this way. You can do some easy algebra to get to the usual y = mx + b form
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Remavas
02/27/2022 17:11:04
And this linear function has a root at x = a
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Hector Salamanca
02/27/2022 17:11:23
Oh yeah that's slop intercept form isn't it
1
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Hector Salamanca
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Remavas
02/27/2022 17:11:32
mhm
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Remavas
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Hector Salamanca
02/27/2022 17:11:36
Yeah
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Remavas
02/27/2022 17:11:48
If we now go to second-degree polynomials
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Remavas
02/27/2022 17:12:19
If we have a 2nd degree polynomial with two roots, we can write it as $y = a(x-b)(x-c)$
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TeXit
02/27/2022 17:12:22
**Remavas**
https://cdn.discordapp.c…980977336321.png
373884980977336321.png
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Hector Salamanca
02/27/2022 17:12:27
Yeah
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Remavas
02/27/2022 17:12:30
where b and c are the roots
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Hector Salamanca
02/27/2022 17:12:38
Si
1
0
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Remavas
02/27/2022 17:12:51
There is a whole theorem on this you can prove with polynomial division, but since you don't know that, we'll resist that urge
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Hector Salamanca
02/27/2022 17:13:19
Thank you 🙏
1
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Remavas
02/27/2022 17:13:28
So what do you think, what will that form look like for a 3rd degree polynomial with 3 real roots?
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Hector Salamanca
02/27/2022 17:13:42
Uh
1
0
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bcfbf534208c4f419b45fafd6d8add8b
603,753,572,412,293,100
Hector Salamanca
02/27/2022 17:14:22
Y = a(x-b)(x-c)(x-d)?
1
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Remavas
02/27/2022 17:14:29
exactly
0
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Hector Salamanca
02/27/2022 17:14:43
Sorry it autocorrected y to yeah for some reason
1
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Remavas
02/27/2022 17:15:10
a happens to be the leading coefficient since after we use FOIL on the product
0
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Remavas
02/27/2022 17:15:18
we'll get $ax^3 + \cdots$
0
1
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bcfbf534208c4f419b45fafd6d8add8b
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TeXit
02/27/2022 17:15:21
**Remavas**
https://cdn.discordapp.c…980977336321.png
373884980977336321.png
0
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Hector Salamanca
02/27/2022 17:15:43
That makes sense yeah
1
0
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Remavas
02/27/2022 17:15:49
and b,c,d are your zeroes
0
1
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Hector Salamanca
02/27/2022 17:16:09
Hm
1
0
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Hector Salamanca
02/27/2022 17:16:18
Alright that makes sense
1
0
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bcfbf534208c4f419b45fafd6d8add8b
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Remavas
02/27/2022 17:18:24
In general, any polynomial can be expressed as $P(x) = a(x-b)^i\cdot (x-c)^j \cdot \cdots \cdot (x^2 + px +q)^l \cdot (x^2 + rx + s)^m \cdot \cdots$
0
1
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help-10
bcfbf534208c4f419b45fafd6d8add8b
510,789,298,321,096,700
TeXit
02/27/2022 17:18:41
**Remavas**
https://cdn.discordapp.c…980977336321.png
373884980977336321.png
0
1
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null
null
null
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948
help-10
bcfbf534208c4f419b45fafd6d8add8b
603,753,572,412,293,100
Hector Salamanca
02/27/2022 17:18:42
Long equation
1
0
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bcfbf534208c4f419b45fafd6d8add8b
373,884,980,977,336,300
Remavas
02/27/2022 17:19:11
Well yeah, but you won't need it for your task
0
1
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Remavas
02/27/2022 17:19:24
this is more than sufficient^
0
1
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bcfbf534208c4f419b45fafd6d8add8b
603,753,572,412,293,100
Hector Salamanca
02/27/2022 17:19:40
Ah thank freyja
1
0
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bcfbf534208c4f419b45fafd6d8add8b
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Remavas
02/27/2022 17:20:35
this also becomes much simpler in the complex plane but that is another story
0
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bcfbf534208c4f419b45fafd6d8add8b
603,753,572,412,293,100
Hector Salamanca
02/27/2022 17:20:57
I'll probably learn that later on in the year lol
1
0
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bcfbf534208c4f419b45fafd6d8add8b
603,753,572,412,293,100
Hector Salamanca
02/27/2022 17:21:00
Or next year
1
0
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bcfbf534208c4f419b45fafd6d8add8b
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Hector Salamanca
02/27/2022 17:22:43
So b would be 14, c would be 1/2, and d would be 9/2
1
0
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Remavas
02/27/2022 17:23:23
yup
0
1
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Hector Salamanca
02/27/2022 17:23:33
Groovy
1
0
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help-10
bcfbf534208c4f419b45fafd6d8add8b
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Hector Salamanca
02/27/2022 17:24:21
So y =(x-14)x(x-1/2)x(x-9/2)
1
0
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bcfbf534208c4f419b45fafd6d8add8b
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Remavas
02/27/2022 17:24:33
you forgot something
0
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Hector Salamanca
02/27/2022 17:24:43
What did I forget?
1
0
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Remavas
02/27/2022 17:24:56
the a
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Hector Salamanca
02/27/2022 17:25:01
Ah yeah
1
0
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Hector Salamanca
02/27/2022 17:25:09
My a key is broken sorry
1
0
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Hector Salamanca
9867
Hector Salamanca
false
948
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Hector Salamanca
02/27/2022 17:25:20
Have to press hella hard
1
0
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Remavas
02/27/2022 17:25:47
well we can replace it with a number directly
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Remavas
02/27/2022 17:25:55
^
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Remavas
02/27/2022 17:26:01
and you're given the leading coefficient
0
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Hector Salamanca
02/27/2022 17:26:25
So a =2
1
0
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Remavas
02/27/2022 17:26:59
And your polynomial becomes $y = 2(x-14)(x-1/2)(x-9/2)$
0
1
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null
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948
help-10
bcfbf534208c4f419b45fafd6d8add8b
510,789,298,321,096,700
TeXit
02/27/2022 17:27:02
**Remavas**
https://cdn.discordapp.c…980977336321.png
373884980977336321.png
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bcfbf534208c4f419b45fafd6d8add8b
603,753,572,412,293,100
Hector Salamanca
02/27/2022 17:27:20
Yeah!
1
0
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bcfbf534208c4f419b45fafd6d8add8b
603,753,572,412,293,100
Hector Salamanca
02/27/2022 17:27:30
And then you have to multiply it, right?
1
0
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Remavas
02/27/2022 17:27:48
Do they ask you to simplify it or put it into a canonical form?
0
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bcfbf534208c4f419b45fafd6d8add8b
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Hector Salamanca
02/27/2022 17:28:08
No but would you mind showing me how anyway?
1
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help-10
bcfbf534208c4f419b45fafd6d8add8b
373,884,980,977,336,300
Remavas
02/27/2022 17:28:38
You just use FOIL
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help-10
bcfbf534208c4f419b45fafd6d8add8b
603,753,572,412,293,100
Hector Salamanca
02/27/2022 17:28:47
What is foil?
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help-10
bcfbf534208c4f419b45fafd6d8add8b
321,445,119,171,756,000
Shuri2060
02/27/2022 17:29:08
$$(a+b)(c+d)$$
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help-10
bcfbf534208c4f419b45fafd6d8add8b
510,789,298,321,096,700
TeXit
02/27/2022 17:29:11
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
bcfbf534208c4f419b45fafd6d8add8b
321,445,119,171,756,000
Shuri2060
02/27/2022 17:29:15
instructions on multiplying this out
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help-10
bcfbf534208c4f419b45fafd6d8add8b
603,753,572,412,293,100
Hector Salamanca
02/27/2022 17:29:33
Ah thank you
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help-10
bcfbf534208c4f419b45fafd6d8add8b
373,884,980,977,336,300
Remavas
02/27/2022 17:29:54
I honestly don't remember myself what the abbreviation exactly means.
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help-10
bcfbf534208c4f419b45fafd6d8add8b
373,884,980,977,336,300
Remavas
02/27/2022 17:30:17
but you should be able to do it^
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help-10
bcfbf534208c4f419b45fafd6d8add8b
603,753,572,412,293,100
Hector Salamanca
02/27/2022 17:30:19
So it is $$(2+14)x(1/2+9/2)$$
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help-10
bcfbf534208c4f419b45fafd6d8add8b
510,789,298,321,096,700
TeXit
02/27/2022 17:30:22
**Modern\_Valkyrie**
https://cdn.discordapp.c…572412293120.png
603753572412293120.png
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help-10
bcfbf534208c4f419b45fafd6d8add8b
373,884,980,977,336,300
Remavas
02/27/2022 17:30:37
woah what what
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help-10
bcfbf534208c4f419b45fafd6d8add8b
373,884,980,977,336,300
Remavas
02/27/2022 17:30:48
We went from a third degree polynomial to a first degree polynomial
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help-10
bcfbf534208c4f419b45fafd6d8add8b
603,753,572,412,293,100
Hector Salamanca
02/27/2022 17:30:49
I'm confused sorry
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help-10
bcfbf534208c4f419b45fafd6d8add8b
373,884,980,977,336,300
Remavas
02/27/2022 17:31:15
https://cdn.discordapp.c…qa_8byvr_704.png
foil_method_math_poster-r4ed14a40e07247af8a06b4108cd83ca6_wqa_8byvr_704.png
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help-10
bcfbf534208c4f419b45fafd6d8add8b
603,753,572,412,293,100
Hector Salamanca
02/27/2022 17:32:28
y = 2(x-14)(x-1/2)(x-9/2)
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help-10
bcfbf534208c4f419b45fafd6d8add8b
321,445,119,171,756,000
Shuri2060
02/27/2022 17:32:49
$$abc = (ab)c = a(bc)$$
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help-10
bcfbf534208c4f419b45fafd6d8add8b
510,789,298,321,096,700
TeXit
02/27/2022 17:32:52
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
bcfbf534208c4f419b45fafd6d8add8b
603,753,572,412,293,100
Hector Salamanca
02/27/2022 17:32:54
I jsut pasted that to read it easier
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help-10
bcfbf534208c4f419b45fafd6d8add8b
603,753,572,412,293,100
Hector Salamanca
02/27/2022 17:33:19
My teacher said that we don't need to simplify btw
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help-10
bcfbf534208c4f419b45fafd6d8add8b
321,445,119,171,756,000
Shuri2060
02/27/2022 17:34:18
I have no idea what the original question was
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help-10
bcfbf534208c4f419b45fafd6d8add8b
321,445,119,171,756,000
Shuri2060
02/27/2022 17:34:40
in which case, yes
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