Example 11 Let the polynomial $h(x)$, when divided by $x-1, x-2, x-$ 3, have remainders $4,8,16$ respectively. Find the remainder when $h(x)$ is divided by $(x-1)(x-2)(x-3)$.
Let $h(x)=(x-1)(x-2)(x-3) q(x)+r(x)$, where $\operatorname{deg} r<3$,

and $r(1)=4, r(2)=8, r(3)=16$. Therefore, by the Lagrange interpolation formula,
$$
\begin{array}{l}
r(x)=\frac{(x-2)(x-3)}{(1-2)(1-3)} \cdot 4+ \\
\frac{(x-1)(x-3)}{(2-1)(2-3)} \cdot 8+\frac{(x-1)(x-2)}{(3-1)(3-2)} \cdot 16 \\
=2\left(x^{2}-5 x+6\right)-8\left(x^{2}-4 x+3\right)+8\left(x^{2}\right. \\
-3 x+2) \\
\quad=2 x^{2}-2 x+4 .
\end{array}
$$