TITLE: An equality relation for complex numbers off the nonnegative real axis
QUESTION [5 upvotes]: For every complex number $z$ off the nonnegative real axis there exist positive numbers $p_0,... ,p_n$ such that $\sum_{i=0}^n p_iz^i = 0$.
Finding difficulty in proceeding with the problem. Need some hints.

REPLY [8 votes]: If $z$ is in the left half plane, there is a quadratic polynomial with non-negative coefficients which has $z$ as a root. If $z$ is not in the left half plane, first find $n$ such that $z^n$ is in the left half plane, then choose a quadratic polynomial in $z^n$.