We consider two polynomial $P(X) =X^{4n}+X^{3n}+X^{2n}+X^{n}+1$ and $Q(X) =X^{4}+X^{3}+X^{2}+X+1$ So what the condition for $P$ devided by $Q$.
we note $P(X)= Q(X) H(x)+ R(X)$ if $P$ devided by $Q$ we should have $R(X)$ since Q is $4$ degree polynomial $R(X)$ should be polynomial of three or less degree .
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Bill Dubuque
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Write $Q(x) = \frac{x^5 - 1}{x - 1}$ and $P(x) = \frac{x^{5n} - 1}{x^n - 1}$. What does this tell you about their roots? – Qiaochu Yuan Sep 27 '23 at 17:58
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Think about the complex roots. – Qiaochu Yuan Sep 27 '23 at 18:07
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If you don't know theorems about roots, you could try using euclidian division? – julio_es_sui_glace Sep 27 '23 at 18:11