Question

If the remainder, when polynomial $f\left(x\right)$ is divided by $x-1,x+1$, are $6,8$ respectively, then the remainder, when $f\left(x\right)$ is divided by$(x-1)(x+1)$ is :

• A. $7-x$
• B. $7+x$
• C. $8-x$
• D. $8+x$

Answer 1

The correct option is A $7-x$

Since the divisor is quadratic, the remainder in general is assumed to be linear.
Thus remainder $=ax+b.$
$\therefore f\left(x\right)=$ Quotient $\times (x-1)(x+1)+$ Remainder

But by remainder theorem,

$f\left(1\right)=6$ and $f(-1)=8$

$\therefore a+b=6$ and $-a+b=8$
$\therefore$ Solving simultaneously, we have

$a=-1$ and $b=7$

$\therefore$ The remainder is $7-x$