Find the common roots of $x^3 + 2x^2 + 2x + 1$ and $x^{1990} + x^{200} + 1$ Find the common roots of $x^3 + 2x^2 + 2x + 1$ and $x^{1990} + x^{200} + 1$.
I completed the first part of the question by finding the roots of the first equation. I obtained $3$ roots, one of them being $-1$ and the other two complex. It is evident that $-1$ is not a root of the second equation, but how can I find out whether the other two roots are common or not?

The roots of $x^2+x+1$ are $\omega$ and $\omega^2$ (The two complex cube roots of unity)
and $\omega^3=1$. Therefore
$$ \omega^{1990}+\omega^{200}+1=\omega+\omega^2+1=0 $$
and
$$ \omega^{2\times1990}+\omega^{2\times200}+1=\omega^2+\omega+1=0$$
So $\omega$ and $\omega^2$ are the common roots.
