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The statement of the problem is:

When polynomial $P(x)$ is divided by $x-\alpha$, the remainder is $\alpha^3$. When it is divided by $x-\beta$, the remainder is $\beta^3$. Find the remainder when $P(x)$ is divided by $(x-\alpha)(x-\beta)$.

I'm not sure where to start.

Arthur
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zazerbayev
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    Well...as a start, write $P(x)=Q(x)\times (x-\alpha)(x-\beta)+R(x)$. What degree is $R(x)$? How many values of $R(x)$ do you need to know to determine it? How many values can you find? – lulu Dec 14 '17 at 16:41
  • I think you need $\alpha\ne \beta$. – lhf Dec 14 '17 at 16:58

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The remainder is of the form $ ax+b$. You need to find $a$ and $b$. So we use the division algorithm : $$P(X)= Q(x) ( x-\alpha)(x- \beta) + ax+b$$. Put in the equation $ x= \alpha, x= \beta$ and solve the simultaneous equation to find $a$ and $b$.

Chirantan Chowdhury
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