Official solution
\[\[\begin{gathered} \text { } \quad z^{1985}+z^{100}+1=0 \\quad z^3+2 z^2+2 z+1=0 \\ (z+1)\left(z^2-z+1\right)+2 z(z+1)=0 \\ (z+1)\left(z^2+z+1\right)=0 \\ \Rightarrow \quad \quad z=-1, \quad z=w, w^2 \\ \text { Now putting } z=-1 \text { not satisfy } \\ \text { Now put } z=w \\ \Rightarrow \quad \quad w^{1985}+w^{100}+1 \\ \Rightarrow \quad \quad w^2+w+1=0 \\ \Rightarrow \quad \quad \text { Also, } z=w^2 \\ \Rightarrow \quad \quad w^{3970}+w^{200}+1 \\ \Rightarrow \quad w+w^2+1=0 \end{gathered}\]\]
Two common root