Is there a polynomial with rational coefficients $P(x)$ such that $\frac {P(n+1)}{P(n)}\in\mathbb{N}$ for every $n\in\mathbb{N}_+$? Does there exist a polynomial with rational coefficients $P(x)$  with the property $\frac {P(n+1)}{P(n)} \in \mathbb{N}$ for every $n \in \mathbb{N}_+$? If so how can I construct such a polynomial?

For every polynomial $P(x)$, one has
$$\lim_{x \rightarrow \infty} \dfrac{P(x+1)}{P(x)}=1,$$
and so if a polynomial satisfies the desired property, one must have $P(n+1)=P(n)$ for all $n$ large enough, hence constant polynomials $P(x)=c$ are the only polynomials with that property. 
