Question

Let $a\ne0$ 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 $-x$.