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 $(x+1)$?

You may use 


*

*$(x^2-1)|(x^{2k} -1)$ by applying $(x^2)^k - 1 = (x^2-1)(x^{2(k-1)} + \cdots + x^2 + 1)$
\begin{eqnarray*} x^{81} + x^{49} + x^{25} + x^{9} + x
& = & x\left((x^2)^{40} -1  + (x^2)^{24} -1  + (x^2)^{12} -1 + (x^2)^{4} -1  + 4+1\right) \\
& = & x((x^2-1)p(x) + 5)\\
& = & x(x^2-1)p(x) + 5x
\end{eqnarray*}
So, the remainder is $5x$.
