Question

If $p\left(x\right)$ is a polynomial of degree greater than $2$ such that $p\left(x\right)$ leaves remainder $a$ and $-a$ when divided by $x+a$ and $x-a$ respectively. If $p\left(x\right)$ is divided by $x^2-a^2$ then remainder is

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

Answer 1

The correct option is D $-x$

As degree of remainder is always lesser than degree of divisor,we have taken remainder with degree less than $2$ here.

Let $ux+v$ be the remainder when $p\left(x\right)$ is divided by $(x^2-a^2)$

Using division algorithm,
$p\left(x\right)=(x^2-a^2)q\left(x\right)+(ux+v)\cdots \left(i\right)$
where $q\left(x\right)$ is the quotient.

We know that, by remainder theorem,
$p(-a)=a,p\left(a\right)=-a$
Putting $x=a$ and $x=-a$ in the equation $\left(i\right)$,
$a=-ua+v-a=ua+v$
Solving them simultaneously,
$\Rightarrow v=0\Rightarrow u=-1$

Hence, the required remainder is
$ux+v=-x+0=-x$