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A polynomial $f(x)$ leaves the remainder 10 when it divided by $x-2$ and the remainder $2x-3$ when divided by $(x+1)^2$. Find the remainder when $f(x)$ is divided by $(x-2)(x+1)^2$.

Attempt

$f(x)=(x-2)q_1(x)+10$, $f(x)=(x+1)^2q_2(x)+(2x-3)$. Then $f(2)=10, f(-1)=-5$. Let the remainder when $f(x)$ is divided by $(x-2)(x+1)^2$ be $ax^2+bx+c.$ then $f(x)=(x-2)(x+1)^2q_3(x)+ax^2+bx+c$

then how to get $a,b,c$. I have only two conditions $f(2)=10, f(-1)=-5$ and three unknowns.

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  • $\begingroup$ Hint: you haven't used the condition that $-1$ is a double root, yet. $\endgroup$ – dxiv Sep 26 '16 at 5:46
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Hint. $-1$ is a double root, so try with the derivative. We have that $f'(x)=2(x+1)q_2(x)+(x+1)^2q'_2(x)+2$, therefore we are able to find also $f'(-1)$.

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