Question 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?