Example 6 The polynomial $x^{4}+m x^{3}+n x-16$ contains the factors $x-1$ and $x-2$. Then $m n=$ $\qquad$
(1989, Sichuan Province Junior High School Mathematics League)
Solution: Let $f(x)=x^{4}+m x^{3}+n x-16$. By the Factor Theorem, we have
$$
\left\{\begin{array} { l } 
{ f ( 1 ) = 0 , } \\
{ f ( 2 ) = 0 , }
\end{array} \text { i.e., } \left\{\begin{array}{l}
m+n=15, \\
8 m+2 n=0 .
\end{array}\right.\right.
$$

Solving, we get $m=-5, n=20$.
Therefore, $m n=-100$.