Find the remainder when you divide $x^{81} + x^{49} + x^{25} + x^{9} + x$ by $x^{3}-x$. Find the remainder when you divide $x^{81} + x^{49} + x^{25} + x^{9} + x$ by $x^{3}-x$.

Attempt:
Let $f(x) = x^{81} + x^{49} + x^{25} + x^{9} + x$, then the remainder of $f(x)$ divided by $(x-1)$ is $f(1)$, divided by $(x+1)$ is $f(-1)$.
Now since $x^{3}-x = x(x^{2}-1)=x(x-1)(x+1)$ is there any relation between the remainder of $f(x)$ divided by $x^{3}-x$ and remainder when $f(x)$ divided by $(x-1)$ and $(x+1)$?

The remainder is a quadratic polynomial $g(x)$ that is divisible by $x$ (since $f(x)$ is divisble by $x$ and has remainders $f(1)$ and $f(-1)$ from division on $(x-1)$ and $(x+1)$ respectively.
One may argue that:
$$
g(x)=f(1)\frac{x+1}{2}+f(-1)\frac{x-1}{-2} + \gamma(x^2-1)
$$
From the fact $g(0) = 0$, one can find the coefficient $\gamma=\frac{f(1)-f(-1)}2$.
