For what values of $n$, where $n$ is an integer, the polynomial $x^3+nx+1$ is reducible over $\Bbb Z$. My attempt:
When $n= 0,-2 $, the given polynomial is reducible over $\Bbb Z$ as $x=-1$ and $x=1$ are zeros of the polynomial. But I couldn't find whether there exists any integer $n$ for which the polynomial $x^3+nx+1$ is reducible over $\Bbb Z$. How can we proceed from here? Is the polynomial irreducible over $\Bbb Z$ if $n$ is not in $\{0,-2\}$?
If it is reducible over $\Bbb{Z}$ then it has a root in $\Bbb{Z}$, say $k\in\Bbb{Z}$. Then $k^3+nk+1=0$ so $$-1=k^3+nk=k(k^2+n),$$ which shows that $k$ divides $-1$, so $k=\pm1$. Solving the two equations $$1^3+n\cdot1+1=0\qquad\text{ and }\qquad (-1)^3+n\cdot(-1)+1=0,$$ yields $n=-2$ and $n=0$ as the only values for which the polynomial is reducible over $\Bbb{Z}$.