Remainder Theorem
If a polynomial \(p(x)\) is divided by \(x-r\), then the remainder is \(p(r)\).
To prove this, let \(p(x)=Q(x)(x-r) + c\). Then let \(x=r\).
Synthetic Division
This works to divide a polynomial \(p(x)\) by \(x-a\) (note that the divisor is monic).
An example best illustrates:
\begin{equation*}
\frac{x^{3}-12 x^{2}-42}{x-3}
\end{equation*}
\begin{equation*}
3 \quad \begin{array}{|lrrr}1 & -12 & 0 & -42 \\ & 3 & -27 & -81 \\ \hline 1 & -9 & -27 & -123\end{array}
\end{equation*}
Thus the result is \(x^{2}-9 x-27\) with a remainder of \(-123\).
Note that synthetic division is also one way to evaluate a polynomial (by virtue of the remainder theorem). It’s also essentially the Horner’s method.