Question 96
Let $\left(P_{n}\right)_{n \in \mathbb{N}}$ be a sequence of polynomials with integer coefficients. Suppose that there exists a monic polynomial $Q$ with integer coefficients such that $P_{n+1}-P_{n}=Q$ for all $n \geqslant 0$. Suppose moreover that for all $n \geqslant 0$, $P_{n}$ has an integer root. Show that we are in one of the following two cases:
- $P_{0}$ and $Q$ have a common integer root;
- there exists a polynomial $R$ with integer coefficients such that $P_{0}=RQ$ and the degree of $R$ is 1.
- $P_{0}$ and $Q$ have a common integer root;
- there exists a polynomial $R$ with integer coefficients such that $P_{0}=RQ$ and the degree of $R$ is 1.