Question

The remainder on dividing the polynomial $q\left(x\right)$ by $(x-a)$ is $k$ and the remainder on dividing the polynomial $r\left(x\right)$ by $(x-a)$ is $-k$.
(a) Find $q\left(a\right)$.
(b) Prove that $(x-a)$ is a factor of the polynomial $q\left(x\right)+r\left(x\right)$.

Answer 1

(a) Dividing $q\left(x\right)$ by $(x-a)$, we get remainder as $k$.
$q\left(a\right)=0$
Dividing $r\left(x\right)$ by $(x-a)$, we get remainder as $-k$.
$r\left(a\right)=0$
$q\left(a\right)+r\left(a\right)=0$
Therefore $q\left(a\right)=-r\left(a\right)$
(b) Let $f\left(x\right)=q\left(x\right)+r\left(x\right)$
$f\left(a\right)=q\left(a\right)+r\left(a\right)=0$
Since $f\left(a\right)=0$, therefore $(x-a)$ is a factor of $f\left(x\right)$.