Roots of Sum of Two Polynomials (with Known Roots)

I am writing a piece of software and I'm trying to avoid root finding polynomials for efficiency purposes. I have two polynomials with complex coefficients, where the roots of both polynomials are known. In fact, the internal representation I use just stores the complex roots and a complex scaling factor, not the polynomial coefficients themselves, although it's trivial to compute these. Essentially, my goal is to sum the two polynomials and express the result in the same internal representation which I use for the addends. The mathematical challenge here is then to find the roots of the sum (and the scaling factor, which is trivial). As I've mentioned, I would very much like to find a way to do this more efficiently than just finding the coefficients of the sum and computing the roots from there - although I will resort to this method as a last resort if what I'm asking for isn't possible.

EDIT: I am putting a bounty on this question, so I want to clarify my standard for a good answer. A good answer will either:

• Explain or overview an algorithm which accomplishes my task, preferably with references to relevant papers with more information.
• Give an explanation for why the task is infeasible, possibly with references or suggestions for another angle of attack, but this is not necessary.

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...

My best guess would be that the way of finding the roots of the sums of polynomials would be a combination of both. You could use the fact that if your 2 polynomials $P \text{ and } Q$ have a common root $z$, then it's also the root of $P+Q$. Then you could use Vieta's formulas (https://en.wikipedia.org/wiki/Vieta's_formulas) to get some information about the sum of your new roots. From there, you could either use trial and error or some other method that I'm not aware of.