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Find the remainder when the polynomial x^5 - 4x^3 + 7x^2 - 3 is divided by (x - 1)(x + 2).  I think there is a way to do this using Remainder Theorem, but I have no idea how.

 Jan 4, 2021
 #1
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We can use long division

 I got 4x+9 but I'm not sure

 

i suggest you search up polynomial division, there are some great tutorials like khan academy that are free

 Jan 5, 2021
 #2
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Remainder theorem will get you there.

 

\(\displaystyle \frac{x^{5}-4x^{3}+7x^{2}-3}{(x-1)(x+2)}=q(x)+\frac{r(x)}{(x-1)(x+2)}.\)

 

q(x) will be a cubic and r(x) will be linear, (one degree lower than the quadratic on the bottom line), so let r(x) = ax + b.

Multiply throughout by (x - 1)(x + 2) to get

 

\(\displaystyle x^{5}-4x^{3}+7x^{2}-3=q(x)(x-1)(x+2)+ax+b\),

 

and now substitute x = 1 and x = -2.

That gets you a pair of simultaneous equations in a and b which you can solve.

 Jan 5, 2021

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