Common roots of the equations $z^3+2z^2+2z+1=0$ and $z^{1985}+z^{100}+1=0$ are

$\begin{array}{l}{\mathrm{z}}^3+2{\mathrm{z}}^2+2\mathrm{z}+1=0\\ \Rightarrow \left({\mathrm{z}}^3+1\right)+2\mathrm{z}(\mathrm{z}+1)=0\\ \Rightarrow (\mathrm{z}+1)\left({\mathrm{z}}^2-\mathrm{z}+1+2\mathrm{z}\right)=0\\ \Rightarrow (\mathrm{z}+1)\left({\mathrm{z}}^2+\mathrm{z}+1\right)=0\\ \Rightarrow \mathrm{z}=-1,\mathrm{\omega },{\mathrm{\omega }}^2\end{array}$

Now equation z¹⁹⁸⁵ + z¹⁰⁰ + 1 = 0 gets satisfied by w and w² but not by –1. So, common roots are $\omega$ and ${\omega }^2$.