Determinants & Inverses · Challenge problem

The determinant $\det\begin{pmatrix}1 & 1 & 1\\a & b & c\\a^3 & b^3 & c^3\end{pmatrix}$ factors as $(b - a)(c - a)(c - b)\,q(a,b,c)$ for a polynomial $q$. Find $q(1, 2, 3)$.