AMO-Bench 2025 Problem 28

For each positive integer $k \ge 2$, define $f(k)$ to be the product of the distinct prime factors of $k$. All the coefficients of the polynomial function $9P(x)$ are rational numbers. Suppose $45 \le P(2025) \le 81$, and there are infinitely many positive integers $n$ for which $P(n) = f(n)$. How many polynomials $P(x)$ satisfy these conditions?