Let be a polynomial of degree greater than 2. If f(1) = 2 and f(2) = 8, find the remainder when f(x) is divided by (x - 1)(x - 2).

Guest Jun 14, 2021

#1**+2 **

Let $q(x)$ be the quotient polynomial and let $r(x)$ be the remainder polynomial when $f(x)$ is divided by $(x-1)(x-2)$. We have

\[

f(x) = q(x)(x-1)(x-2) + r(x).

\]

Since $(x-1)(x-2)$ is quadratic, we can write $r(x)$ as $ax+b$ for some constants $a$, $b$. From $f(1)=2$ and $f(2)=8$ we get

\begin{eqnarray*}

2 = f(1) = r(1) = a+b \\

8 = f(2) = r(2) = 2a+b

\end{eqnarray*}

Solving, we get $a = 6$ and $b = -4$, so the remainder $r(x)=6x-4$.

Bginner Jun 14, 2021