Polynomial $P(x)$ gives a remainder of 2 when divided by $x-1$, and a remainder of 1 when divided by $x-2$.

What remainder does $P(x)$ give when divided by the polynomial $(x-1)(x-2)$?

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Let $P(x)=(x-1)(x-2) Q(x)+a x+b$. By the Remainder Theorem, $a+b=P(1)=2, 2a+b=P(2)=1$, from which $a=-1, b=3$.

## Answer

$3-x$.