<s>[INST] 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?
 [/INST] The solution to the equation $x^2+x+1=0$  is  $e^{\frac{2\pi}{3}}$ and $e^{\frac{4\pi}{3}}$ .Having $1990=3\times 663 +1$ and $200=3\times 66 +2 $, we can evaluate the second equation as 
if $ x= e^{\frac{2\pi}{3}} $ then $x^{1990}=x^{3\times 663 +1}=x,x^{200}=x^{3\times 66 +2}=x^2$, so $e^{\frac{2\pi}{3}}$ is root of second equation.
by the property of real polynomials, $\textit{conjugate of a root is also a root of that polynomial}$ . 
So, the two complex solution of the first equation is also solution for the second one.
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