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 $\omega^2$. 

$$x^3+2x^2+2x+1=(x+1)(x^2+x+1)$$
The roots are $-1,\omega,$ and $\omega^2$, where $\omega,\omega^2$ are non real cube roots of unity.
Substituting in the other equation,
$$(-1)^{200}+(-1)^{130}+1=1\ne0$$
$$\omega^{200}+\omega^{130}+1=\omega^2+\omega+1=0$$
$$(\omega^2)^{200}+(\omega^2)^{130}+1=\omega+\omega^2+1=0$$
Thus, two roots are common.
