Question 159
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
$$
Q(x)^{2n} - 1 = R_n(x) (P(x)^{2n} - 1).
$$
$$
Q(x)^{2n} - 1 = R_n(x) (P(x)^{2n} - 1).
$$