Finding all complex roots of $z^{11}+z=1$ with $|z|=1$

Question

Find all complex roots of $z^{11}+z=1$ with $|z|=1$.

I have factored the equation into $$(z^2-z+1)(z^9+z^8-z^6-z^5+z^3+z^2-1)=0$$ and found the roots to the first part: ${1\over 2} + {i\sqrt{3}\over 2}$ and ${1\over 2} - {i\sqrt{3}\over 2}$ but I do not know how to proceed.

Answer 1

Since $z^{10}+1 ={1\over z}$ we have $$|z^{10}+1 | = 1$$ so $z^{10}$ is on circle with center at $-1$ and radius $1$. But $z^{10}$ is also on a circle with center at $0$ and radius $1$. So $$z^{10} = \cos {2\pi \over 3}\pm i\sin {2\pi \over 3}$$ and now should be easy...