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Let $a \neq 0$ 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 $x-a$, then the remainder when $p(x)$ is divided by $x^2-a^2$ is :
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Verified Answer
The correct answer is:
$-x$
Let the remainder be $R(x)$, then
$R(x)=P(x)+q$
Given, $\quad R(-a)=a \Rightarrow-P a+q=a$ $\ldots$ (i)
and $R(a)=-a \Rightarrow P a+q=-a$$\ldots$ (ii)
On solving Eqs. (i) and (ii), we get
$P=1, q=0$
$\therefore \quad R(x)=-x$
$R(x)=P(x)+q$
Given, $\quad R(-a)=a \Rightarrow-P a+q=a$ $\ldots$ (i)
and $R(a)=-a \Rightarrow P a+q=-a$$\ldots$ (ii)
On solving Eqs. (i) and (ii), we get
$P=1, q=0$
$\therefore \quad R(x)=-x$
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