Question

If the remainders of the polynomial f(x) when divided by x+1 and x-1 are 3, 7 then the remainder of f(x) when divided by $(x^2-1)$ is

• A. x + 4
• B. 2x + 3
• C. 2x + 4
• D. 2x + 5

Answer 1

The correct option is D 2x + 5

According to remainder theorem

$f\left(x\right)=Q_1\left(x\right)(x^2-1)+r\left(x\right)$

$r\left(x\right)=ax+b$
So, $f\left(x\right)=Q_1\left(x\right)(x^2-1)+(ax+b)$

$f(-1)=-a+b=3$ $\dots \left(1\right)$

$f\left(1\right)=a+b=7$ $\dots \left(2\right)$

Add $\left(1\right)$ and $\left(2\right)$

$-a+b+a+b=10\Rightarrow 2b=10\Rightarrow b=5$

substitute it in $\left(1\right)$ then

$a=5-b=5-3=2$

$\Rightarrow b=5;a=2$

So $r\left(x\right)=2x+5$