Question 25
We call a sequence $(P_n)_{n=1,...}$ of polynomials an *arithmetic sequence with common difference* $Q(x)$ if $P_{n+1} = P_n + Q$, $n = 1, \dots$. Suppose that we have an arithmetic sequence of polynomials with the common difference $Q(x)$ and the first term $P(x)$ such that $P$, $Q$ are *monic* polynomials with integer coefficients and have no common root. Further, each element of this arithmetic sequence has at least one integer root. Prove that
i. $Q(x)$ divides $P(x)$;
ii. The polynomial $P(x)/Q(x)$ is of degree one.
i. $Q(x)$ divides $P(x)$;
ii. The polynomial $P(x)/Q(x)$ is of degree one.