How to show $x^3 - x^2 -11x-12$ is irreducible in $\mathbb{Q}[x]$

Question

How can I show that the polynomial $f(x)=x^3 - x^2 -11x-12$ is irreducible in $\mathbb{Q}[x]$, I've tried Einstein's criterion but it fails. I'm not sure about any other methos I could use to show this.

Answer 1

If it were reducible it would have a linear factor, and then an integer solution (observe it is monic). We can check the candidates for integers solutions, which must be divisors of $12$.

Check if $\pm1,\pm2,\pm3,\pm4,\pm6,\pm12$ are solutions. If none of them is a solution then it must be irreducible over $\mathbb{Q}$.

Answer 2

The polynomial $f(x)$ is irreducible if and only if $f(x+6)$ is irreducible. In this case $$f(x+6)=x^3+17x^2+85x+102\ ,$$ which is irreducible by Eisenstein with $p=17$.