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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).

 Jun 14, 2021
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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 $r(x)=6x-4$.

 Jun 14, 2021

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