How to show that $4x^9 -9x^3 + 24x + 13 $ is irreducible over $\mathbb{Q}$

Question

How to show that

$4x^9 -9x^3 + 24x + 13 $ is irreducible over $\mathbb{Q}$.

Since the polynomial is primitive over $\mathbb{Z}$, hence I can show instead that it is irreducible over $\mathbb{Z}$.

I'm thinking of using $\mod p$ irreducibility test here, but it's still hard to prove it since it's in degree 9. Meaning I will have to check individually that it does not have linear, quadratic, etc. factors.

Do you know of any easier way to prove this?

Answer 1

Use $x=y-1$ and Eisenstein for $p=3$.

Since $$-4+9-24+13=-6$$ is not divided by $9$, we are done!