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?

$$x^3+2x^2+2x+1=x^3+x^2+x^2+x+x+1=(x+1)(x^2+x+1).$$
We see that $-1$ it's not common root and
$$x^{1990}+x^{200}+1=x^{1990}-x+x^{200}-x^2+x^2+x+1=$$
$$=x((x^3)^{663}-1)+x^2((x^3)^{66}-1)+x^2+x+1\equiv0(\mod(x^2+x+1)),$$
which says that roots of $x^2+x+1$ are all common roots and we get the answer:
$$\left\{-\frac{1}{2}+\frac{\sqrt3}{2}i,-\frac{1}{2}-\frac{\sqrt3}{2}i\right\}$$
