1. (1) If $z^{2}+z+1=0$, find the value of $z^{2005}+z^{2006}+z^{2008}+z^{2009}$;
(2) If the imaginary number $z$ satisfies $z^{3}=8$, find the value of $z^{3}+z^{2}+2 z+2$.
1. (1) From $z^{2}+z+1=0$, we get $z^{3}=1$ and $z \neq 1$, so $z^{2105}+z^{2006}+z^{2008}+z^{2009}=2\left(z+z^{2}\right)=-2$.
(2) From the given, we have $z=2 w$, where $w$ is a complex cube root of unity. Hence $z^{3}+z^{2}+2 z+2=8 w^{3}+4 w^{2}+4 w+2=$ $10+4(w+w)=6$.