the common roots of the equation z^3+2z^2+2z+1=0 and z^1985 + z^100 + 1=0 are​

Dear Student,

To find the common roots of $z^3+2z^2+2z+1=0$ and $z^{1985}+z^{100}+1=0$, first let us find roots of $z^3+2z^2+2z+1=0$.
$$\begin{gathered} z^3+2z^2+2z+1=0 \\ \Rightarrow z^3+1+2z\left(z+1\right)=0 \\ \Rightarrow z^3+1^3+2z\left(z+1\right)=0 \\ \Rightarrow \left(z+1\right)\left(z^2-z+1\right)+2z\left(z+1\right)=0 \\ \Rightarrow \left(z+1\right)\left(z^2-z+1+2z\right)=0 \\ \Rightarrow \left(z+1\right)\left(z^2+z+1\right)=0 \\ z+1=0 \Rightarrow z=-1 \\ \left(z^2+z+1\right)=0 \Rightarrow z=\omega , {\omega }^2 \end{gathered}$$

Hence, roots of $z^3+2z^2+2z+1=0$ are $-1, \omega , {\omega }^2$.
$$\begin{gathered} For, z=-1 \\ {z}^{1985}+z^{100}+1={\left(-1\right)}^{1985}+{\left(-1\right)}^{100}+1=-1+1+1=1\ne 0 \\ For, z=\omega \\ {z}^{1985}+z^{100}+1={\left(\omega \right)}^{1985}+{\left(\omega \right)}^{100}+1={\omega }^2+\omega +1=0 \left[{\omega }^3=1 and {\omega }^2+\omega +1=0\right] \\ For, z={\omega }^2 \\ {z}^{1985}+z^{100}+1={\left({\omega }^2\right)}^{1985}+{\left({\omega }^2\right)}^{100}+1={\omega }^{3970}+{\omega }^{200}+1=\omega +{\omega }^2+1=0 \end{gathered}$$

Hence, roots of $z^{1985}+z^{100}+1=0$ are $\omega , {\omega }^2$.
Thus, the common roots of $z^3+2z^2+2z+1=0$ and $z^{1985}+z^{100}+1=0$ are $\omega , {\omega }^2$.
Regards,