Find $\sum_{k=1}^n P(z_k)$ where P is a polynomial and $z_k$ the $k$th root of $z^{13}=1$ Question: Given a complex number z such that $z^{13}=1$, find the sum of all possible values of $z+z^3+z^4+z^9+z^{10}+z^{12}$.
I know we have to use roots of unity and try to manipulate the polynomial we want to evaluate, but can't find a pattern.

As stated in comments, for a fixed $z=e^{i\theta}$ the given expression is equal to $2(\cos\theta+\cos3\theta+\cos4\theta)$. As $z$ runs through its $13$ possible values, so does each term ($1,3,4$ are all coprime to $13$) and the sum over all possibilities is
$$6\sum_{k=0}^{12}\cos\frac{2\pi k}{13}$$
but the summation is well-known to evaluate to zero, so the final answer is $0$.
