Roots of Sum of Two Polynomials (with Known Roots)

There is no nice formula to get the roots of $P+Q$ from the roots of $P$ and of $Q$. For example, the roots of $x^5$ and $2x+1$ are easy to find, but the sum of these polynomials is $x^5 + 2 x + 1$, an irreducible quintic whose roots can't be expressed in radicals.

EDIT: In the case of a polynomial with three or four terms, we can express it as the sum of two polynomials with one or two terms, and then there is only trivial "additional information about the polynomials". Any "analytic method" is going to have to find roots of polynomials with three or four terms. And then once you have those roots, you can apply your "method" again to find roots of polynomials with five to eight terms...