Remainders of polynomials of higher degrees. Let $R(x)$ be a remainder upon dividing $x^{44}+x^{33}+x^{22}+x^{11} +1$ by the polynomial $x^4 +x^3 +x^2 +x +1$. Find: $R(1)+2R(2)+3R(3)$. Answer provided is $0$

The thing that makes this easy is $x^4 + x^3 + x^2 + x + 1$ divides $x^5 - 1$. Thus, if
$$\begin{align} f(x) &\equiv g(x) & &\pmod{x^5 - 1} 
\\ g(x) &\equiv h(x) & &\pmod{x^4 + x^3 + x^2 + x + 1} 
\end{align}$$
then
$$ f(x) \equiv h(x) \pmod{x^4 + x^3 + x^2 + x + 1} $$
and it's easy to find a low degree $g(x)$ to use as the intermediate, since $x^5 \equiv 1 \pmod{x^5-1}$.

But even without that, you can use the usual tricks for modular arithmetic; e.g. some of the content of How do I compute $a^b\,\bmod c$ by hand? applies here too.
