2021 Fall AMC 12B Problem 14

Suppose that P(z), Q(z), and R(z) are polynomials with real coefficients, having degrees 2, 3, and 6, respectively, and constant terms 1, 2, and 3, respectively. Let N be the number of distinct complex numbers z that satisfy the equation P(z) \cdot Q(z)=R(z). What is the minimum possible value of N?

\textbf{(A)}\: 0\qquad\textbf{(B)} \: 1\qquad\textbf{(C)} \: 2\qquad\textbf{(D)} \: 3\qquad\textbf{(E)} \: 5

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.