TITLE: roots of sum of two polynomials
QUESTION [10 upvotes]: 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 $A$?

REPLY [9 votes]: If they are complex polynomials or can be treated as such, then you could apply Rouche's 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 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.  There are a few papers which explore this specifically

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