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A polynomial $f(x)$ gives remainder $2$ when divided by $(x-3)$ and gives a remainder $-4$ when divided by $(x+1)$. What is the remainder when $f(x)$ is divided by $(x^2 - 2x - 3)$?

I have shortened the question by:

1.Showing that $f(-1) = -4$ and $f(3) = 2$ by remainder theorem respectively.

2.Figured out that $(x^2 - 2x - 3)= (x-1)(x-3)$.

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You are on the right track.

Hint: For some polynomial $s(x),$ and for some $a,b \in \mathbb{R},$

$f(x) = (x^2 - 2x - 3) \cdot s(x)+(ax+b) = (x+1)(x-3) \cdot s(x)+(ax+b).$

Can you see why?

How then can you use this and everything else you know about $f$?

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