If they are complex polynomials or can be treated as such, then you could apply [Rouche's theorem][1], 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][2] 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][3].  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.*


  [1]: http://en.wikipedia.org/wiki/Rouch%25C3%25A9%2527s_theorem
  [2]: http://en.wikipedia.org/wiki/Wronskian
  [3]: http://mathworld.wolfram.com/MasonsTheorem.html