Question

Let $a\ne 0$ and $p\left(x\right)$ be a polynomial of degree greater than $2$. If p(x) leaves remainders $a$ and $-a$ when divided respectively, by $x+a$ and $x-a$, 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 $-x$

We are given that $p(-a)=a$ and $p\left(a\right)=-a$
[When a polynomial $f\left(x\right)$ is divided by $x-a$ , remainder is $f\left(a\right)$].
Let the remainder, when $p\left(x\right)$ is divided by $x^2-a^2$, be $Ax+B$. Then,
$p=\left(x\right)=Q\left(x\right)(x^2-a^2)+Ax+B$ (1)
where $Q\left(x\right)$ is the quotient. Putting $x=a$ and $-a$ in (1), we get
$p\left(a\right)=0+Aa+B\Rightarrow -a=Aa+B$ (2)
and $p(-a)=0-aA+B\Rightarrow a=-aA+B$ (3)
Solving (2) and (3), we get
$B=0$ and $A=-1$
Hence, the required remainder is $-x.$