Is there any special relationship between the roots of $P(x)$ and $P(x+q) \, \,\text{where}\,q\in\mathbb{Z}$? $P(x)$ is an arbitrary polynomial with an arbitrary degree.
I've notice something. When $P(x+q)$ is divided by $(x+q)$, the remainder is always $$r(x)=r_1r_2r_3\cdots r_n=\prod_{i=1}^{n}r_i\,\text{where $r_i$ are the roots of $P(x)$}$$ Is this just arbitrary?