On the largest prime divisor of polynomial and related problem

Abstract: We denote $\mathcal{P}$ = ${P(x)|$ $P(n) \mid n!$ for infinitely many $n}$. This article identifies some polynomials that belong to $\mathcal{P}$. Additionally, we also denote $P^+(m)$ as the largest prime factor of $m$. Then, a consequence of this work shows that there are infinitely many $n \in \mathbb{N}$ so that $P^+(f(n)) < n^{\frac{3}{4}+\varepsilon}$ if $f(x)$ is cubic polynomial, $P^+(f(n)) < n$ if $f(x)$ is reducible quartic polynomial and $P^+(f(n)) < n^{\varepsilon}$ if $f(x)$ is Chebyshev polynomial.