Why is $x^3+2x^2+x-9$ irreducible over $\mathbb{Q}$?

Question

Eisenstein's criterion doesn't apply directly to this polynomial, so I have been looking for substitutions of the form $x+a$, with $a \in \mathbb{Q}$, in order to use Eisenstein's criterion on a 'new' polynomial (If this new polynomial is irreducible over $\mathbb{Q}$, then the original one in question must also be, because a substitution is an automorphism).

Maybe there is a different method entirely. Any help would be appreciated!

Answer 1

Hint: It is a cubic polynomial, therefore if it wasn't irreducible it must have a root. Now use the rational root test.

Answer 2

1. $x^3+x+1$ is an irreducible polynomial over $\mathbb{F}_2$
2. Your polynomial is irreducible over $\mathbb{F}_2$
3. Your polynomial is irreducible over $\mathbb{Q}$.