Example 4 Find the remainder when $f(x)=x^{81}+x^{49}+x^{25}+x^{9}+x$ is divided by $g(x)=x^{3}-x$.
Solve: From $g(x)=x^{3}-x=x(x-1)(x+1)$, we know $x_{1}=0, x_{2}=1, x_{3}=-1$.
At this point, $f(0)=0, f(1)=5, f(-1)=-5$.
$$
\begin{array}{l}
r(x)=\frac{(x-1)(x+1)}{(0-1)(0+1)} \cdot f(0)+\frac{(x-0)(x+1)}{(1-0)(1+1)} \cdot f(x)+\frac{(x-0)(x-1)}{(-1-0)(-1-1)} \cdot f(-1) \\
=\frac{5 x(x+1)}{1 \cdot 2}+\frac{-5 x(x-1)}{(-1)(-2)}=5 x. \\
\end{array}
$$

Therefore, the remainder when $f(x)=x^{81}+x^{49}+x^{25}+x^{9}+x$ is divided by $g(x)=x^{3}-x$ is $r(x)=5 x$.