2025 AMC 12A Problem 25

Polynomials P(x) and Q(x) each have degree 3 and leading coefficient 1, and their roots are all elements of \{1, 2, 3, 4, 5\}. The function f(x) = \dfrac{P(x)}{Q(x)} has the property that there exist real numbers a \lt b \lt c \lt d such that the set of all real numbers x such that f(x) \le 0 consists of the closed interval [a, b] together with the open interval (c, d). How many functions f(x) are possible?

\textbf{(A)} 7 \qquad \textbf{(B)} 9 \qquad \textbf{(C)} 11 \qquad \textbf{(D)} 12 \qquad \textbf{(E)} 13

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