For which values of $k$ does the polynomial $X^{2017}-X^{2016}+X^{2}+k X+1$ have a rational root?
Our last lemma is essential to solve this exercise. If $x=\frac{r}{s}$ is a rational root of $X^{2017}-X^{2016}+X^{2}+k X+1$, then $r \mid 1$ and $s \mid 1$. Thus $x \in\{-1,1\}$. By substitution, for 1 to be a root, it is necessary that $k=-2$ and for -1 to be a root, it is necessary that $k=0$.