Q107 / 160 · TS EAMCET 2003 · Mathematics

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 :

  • A
    $x$
  • Correct answer
    B
    $-x$
  • C
    $-2 x$
  • D
    $2 x$

Solution

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 $P=1, q=0$ $\therefore \quad R(x)=-x$

Hence, the correct answer is option: 'B'.

Relevant Questions