Question

Let $a\ne 0$ and $p(x)$ 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 $p(x)$ is divided by $x^2-a^2$ is

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

Answer 1

Answer: (D)

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