Problem

Source: Vietnam TST 2024 P6

Tags: algebra, polynomial



Let $P(x) \in \mathbb{Z}[x]$ be a polynomial. Determine all polynomials $Q(x) \in \mathbb{Z}[x]$, such that for every positive integer $n$, there exists a polynomial $R_n(x) \in \mathbb{Z}[x]$ satisfies $$Q(x)^{2n} - 1 = R_n(x)\left(P(x)^{2n} - 1\right).$$