A Polynomial Based Counting Puzzle

Let $w \in \mathbb{C}$ . We define $A(w)$ to be the set of all non-constant polynomials $P$ such that the following properties hold:

• If $r$ is a root of $P$ , then so is $wr$ .
• All roots of $P$ have a multiplicity of 1.
• All roots of $P$ are non-zero.

Let $Q_1, Q_2$ be polynomials of degree 8 and 12 respectively, where $Q_1$ and $Q_2$ do not share any roots. Determine the number of ordered triples $(w_1, w_2, w_3)$ there are such that if $Q_1 \in A(w_1)$ and $Q_2 \in A(w_2)$ (provided such $Q_1, Q_2$ exist) then $Q_1 Q_2 \in A(w_3)$ .