Question 31

Explanation

Given that $f(x)$ is divisible by $(x - 1)$, then $f(1) = 0$.

We are asked to find the remainder when $f(x)$ is divided by $(x - 1)(x + 1)$.

The formula used is if $f(x)$ is divided by $(x - a)(x - b)$, then the remainder is

Therefore, when $f(x)$ is divided by $(x - 1)(x + 1)$, the remainder is

Since $f(1) = 0$, then

Therefore, the remainder when $f(x)$ is divided by $(x - 1)(x + 1)$ is $\frac{f(-1)}{2}(1 - x)$.