Review question

# What if dividing $f(x)$ by $(x-a)(x-b)$ and by $(x-a)(x-c)$ gives the same remainder? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8682

## Question

Show that the remainder when the polynomial $f(x)$ is divided by $(x-a)$ is $f(a)$. Show further that, if $f(x)$ is divided by $(x-a)(x-b)$, where $a \ne b$, then the remainder is $\begin{equation*} \left( \frac{f(a) - f(b)}{a-b} \right)x + \left( \frac{af(b) - bf(a)}{a-b} \right). \end{equation*}$ The remainders when $f(x)$ is divided by $(x-a)(x-b)$ and by $(x-a)(x-c)$, $a \ne b \ne c$, are equal. Prove that $\begin{equation*} (b-c)f(a) + (c-a)f(b) + (a-b)f(c) = 0. \end{equation*}$