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

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.

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

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.