$n,a\in \mathbb Z,n\geq1,$ prove that $x^3+x+1\nmid x^n+a$

Question

$n,a\in \mathbb Z,n\geq1,$ prove that $x^3+x+1\nmid x^n+a.$ In other word, they have no common roots.

My idea: Let $x_1,x_2,x_3$ be the roots of $x^3+x+1=0,$ we need to prove that $\dfrac{x_1}{x_2}$ is not a root of unity.

Answer 1

Hint: if $x^3 + x + 1 \mid x^n + a$, any root of $x^3 + x + 1$ is a root of $x^n + a$. Consider the real root.

Answer 2

Hint: We take your approach a little further. Since the ratios are roots of unity, the roots have equal norm. The norm of the product is $1$, so the roots have norm $1$.