I believe that there is no common theory for finding roots of polynomial sum. In my case I have 
$$P\_{n}(x)+AQ\_{n}(x)$$.
I am wondering how roots of this sum depend on $A$?



roots of sum of two polynomials
If they are complex polynomials or can be treated as such, then you could apply [Rouche's theorem](http://en.wikipedia.org/wiki/Rouch%C3%A9%27s_theorem), where the location of the zeros is determined by the dominant polynomial within the sum. (*"Walk the dog on the leash"*)


Possibly related: you could use the [Wronskian](http://en.wikipedia.org/wiki/Wronskian) to determine the values of A that make $P\_n(x)$ and $Q\_n(x)$ linearly independent.


Your question is related to [Mason's theorem](http://mathworld.wolfram.com/MasonsTheorem.html). There are a few papers which explore this specifically


1. *MR1923392 (2003j:30012) Kim, Seon-Hong . Factorization of sums of
polynomials. Acta Appl. Math. 73 
(2002), no. 3, 275--284.*
2. *MR2103113 (2005h:30011) Kim,
Seon-Hong . On zeros of certain sums
of polynomials. Bull. Korean Math.
Soc. 41 (2004), no. 4, 641--646*
3. *MR1911767 (2003d:11036) Pintér, Á. 
Zeros of the sum of polynomials. J.
Math. Anal. Appl. 270 (2002), no.
1, 303--305.*


