<s>[INST] Polynomial factors Why must $x^2 + x + 1$ be a factor of  $x^5+x^4+x^3+x^2+x+1$?
I know that when we divide  $x^5+x^4+x^3+x^2+x+1$ by $x^3+1$ we get $x^2 + x + 1$, but is there an argument/theorem or anything that could tell that $x^2+x+1$ must divide $x^2 + x + 1$?
 [/INST] $(x^2+x+1)(x-1)=x^3-1$ and $(x^5+x^4+x^3+x^2+x+1)(x-1)=x^6-1$. Then note that $x^6-1=(x^3-1)(x^3+1)$. Thus in effect it is the third binmoial formula.
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