<s>[INST] Number of common roots of $x^3 + 2 x^2 +2x +1 = 0$ and $x^{200} + x^{130} + 1 = 0 $ The equations $x^3 + 2 x^2 +2x +1 = 0$ and $x^{200} + x^{130} + 1 = 0 $ have   


*

*exactly one common root;

*no common root;

*exactly three common roots;

*exactly two common roots.


I factored the first equation. I think the roots are $-1$, $\omega$ and $\omega^2$. 
 [/INST] $\gcd (x^3 + 2 x^2 +2x +1 ,  x^{200} + x^{130} + 1 ) = x^2+x+1$, so two roots.
Note that you don't need to know the two roots, just that the gcd has degree $2$ and is coprime with its derivative (so no repeated roots).
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