2. Find the remainder when $f(x)=x^{81}+x^{49}+x^{25}+x^{9}+x$ is divided by $g(x)=x^{3}-x$.
2. 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, $f(0)=0, f(1)=5$. $f(-1)=-5$.
Let the remainder of $f(x)$ divided by $g(x)$ be $r(x)$, then $\operatorname{deg} r<3$. By the Lagrange interpolation formula, we get
$$
\begin{aligned}
r(x) & =\frac{(x-1)(x+1)}{(0-1)(0+1)} f(0)+\frac{(x-0)(x+1)}{(1-0)(1+1)} f(1)+\frac{(x-0)(x-1)}{(-1-0)(-1-1)} f(-1) \\
& =5 x .
\end{aligned}
$$