Find all numbers $n\geq1$ for which the polynomial $x^{n+1}+x^n+1$ is divisible by
(a) $x^2+x+1$
(b) $x^2-x+1$.
I tried to do long division, but I didn't really seem to be going anywhere. Any suggestions? thanks
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Please include what you've tried in attempting to solve the problem. Perhaps playing around with long division might help? It's still doable with three terms. https://en.wikipedia.org/wiki/Polynomial_long_division – Shuri2060 Jul 06 '17 at 23:51
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[How to ask a good question?](https://math.meta.stackexchange.com/questions/9959/how-to-ask-a-good-question) – Sahiba Arora Jul 06 '17 at 23:51
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Hint $ $ mod $\,x^2+x+1\!:\,\ x^3\equiv 1\,\Rightarrow\, x^n\equiv x^{n\bmod 3}$ and similarly for the other (negate $x)$
Bill Dubuque
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[See here](https://math.stackexchange.com/a/2012356/242) for some numerical examples. – Bill Dubuque Jul 07 '17 at 00:13
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Hint:
It is divisible by $x^2+x+1$ if and only the (complex) roots of $x^2+x+1$ are roots of $x^{n+1}+x^n+1$. Now the complex roots of the quadratic polynomial are the non-real cube roots of unity, $j$ and $j^2$. Calculate the successive powers of $j$ and plug them in $x^{n+1}+x^n+1$ to check.
Similarly, the roots of $x^2-x+1$ are the cube roots of $-1$, namely $-j$ and $-j^2$.
Bernard
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