Let $P,Q$ be polynomials (with whatever coefficients you desire.) In general, the roots of $P+Q$ are not immediately calculable (except by direct computation) even if we have full information of $P$ and $Q$, including their roots.
My question: what are some non-trivial* theorems that give information on the roots of $P+Q$ given information on $P$ and $Q$, probably with certain restrictions? Ie. their location, sum, exact value(s)...anything!
Example: One theorem which fits the bill is Rouché's theorem, which can be applied to polynomials.
This question is, partly, a reference request.
*By non-trivial, I mean $P \neq-Q$, or things of that nature.
I am especially interested in the case that $P,Q$ both have roots strictly on the unit circle, and/or $P,Q$ restricted to $\mathbb{Z}[x]$.