<s>[INST] 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 $0$
 [/INST] $$x^{44}+x^{33}+x^{22}+x^{11}+1=\frac{x^{55}-1}{x^{11}-1}=\frac{(x^5-1)\sum_{k=0}^{10}x^{5k}}{x^{11}-1}\\=\frac{x^5-1}{x-1}\cdot\frac{\sum_{k=0}^{10}x^{5k}}{\sum_{k=0}^{10}x^k}=(x^4+x^3+x^2+x+1)\cdot\color{blue}{\frac{\sum_{k=0}^{10}x^{5k}}{\sum_{k=0}^{10}x^k}}$$ 
The blue fraction is actually a polynomial because the only eleventh root of $1$ which is also a fifth root of $1$ is $1$ itself. Or, if you prefer, because $\operatorname{gcd}(x^{11}-1,x^5-1)$ is obviously $x-1$.
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