Find all values of $k$ such that $x^2+x+1$ is a factor of $x^{2k}+x^k+1$. I tried treating the first polynomial as a root of the other but didn’t get anywhere :(. I also tried substitution to get the second polynomial the resemble the first one but also didn’t get anywhere. Just $k=1$?
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I don't know what you mean by "treating the first polynomial as a root of the other". Hint: the two roots of the first one must be roots of the second one. – Anne Bauval Feb 09 '24 at 14:45
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Please format your formulas. – Anne Bauval Feb 09 '24 at 14:47
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Let $\omega$ be a root of $x^2+x+1$, then write $$x^{2k}+x^k+1=(x^k)^2+(x^k)+1$$
The task has now been reduced to find out for which $k$ is $\omega^k$ a root of $$x^2+x+1$$
Hint:
What is the value of $\omega^3$?
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