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Completley impromptu, one of my extended middle school students asked a question about her additional maths she was studying outside of school.

For a certain polynomial, f(x), the remainder on division by $(x-1)^2$ is $(2x-1)$, and $f(x)$ leaves a remainder of $1$ on division by $(x-3)$. Find the remainder on division by $(x-1)^2(x-3)$.

Any ideas?

Cliff
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1 Answers1

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Let $f(x)=g(x)(x-1)^2(x-3)+A(x-1)^2+B(x-3)+C$

Using Polynomial remainder theorem, $1=f(3)=4A+C\iff C=-4A$

and $f(x)\equiv Bx+C-3B\pmod{(x-1)^2}\implies B=2,C-3B=-1\iff C=?$

lab bhattacharjee
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