Question

Let $a\ne 0$ and $P\left(x\right)$ be a polynomial of degree greater than $2$, If $P\left(x\right)$ leaves remainders $a$ and $-a$ when divided respectively by $x+a$ and $x-a$, then the remainder when $P\left(x\right)$ is divided by $x^2-a^2$, is

• A. $2x$
• B. $-2x$
• C. $x$
• D. $-x$

Answer 1

The correct option is D $-x$

Let $P\left(x\right)=(x-\alpha )(x+\alpha )Q\left(x\right)+rx+s$
When divided by $x+\alpha$, we get
$P(-\alpha )=(-\alpha -\alpha )(-\alpha +\alpha )Q(-\alpha )+r(-\alpha )+s=\alpha$
$\Rightarrow s-r\alpha =\alpha$ ...(1)
And when divided by $x-\alpha$, we get
$P\left(\alpha \right)=(\alpha -\alpha )(\alpha +\alpha )Q\left(\alpha \right)+r\alpha +s=-\alpha$
$\Rightarrow s+r\alpha =-\alpha$ ...(2)
Solving (1) and (2), we get
$s=0,r=-1$
Hence, $P\left(x\right)=(x-\alpha )(x+\alpha )Q\left(x\right)-x$
And when divided by $x^2-{\alpha }^2$ leaves remainder $-x$.