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?

$$x^2+x+1=0$$
Let's just consider$$x=\exp\left(\frac{i\pi}{3}\right).$$
$$1990 \equiv 1989+1 \equiv 1 \mod 3$$
$$200 \equiv 198+2 \equiv 2 \mod 3$$
Hence $$x^{1990}+x^{200}+1 = x+x^2+1=0$$
$\exp\left(-\frac{i\pi}{3} \right)$ being the conjugate must be another solution as well.
