Is the polynomial $\frac{1}{64}x^6 + 3x^4 - \frac{1}{4}x^3 - x + 6$ irreducible over $\Bbb Q$?

Question

How do I approach proving irreducibility/reducibility in a polynomial with rational coefficients? Can I apply Eisenstein in some way?

Is the polynomial $\frac{1}{64}x^6 + 3x^4 - \frac{1}{4}x^3 - x + 6$ irreducible over $\Bbb Q$?

Answer 1

Replace $x$ by $2y$ gives $$\frac{1}{64}x^6 + 3x^4 - \frac{1}{4}x^3 - x + 6 = y^6+48y^4-2y^3-2y+6$$ Now apply Eisenstein criterion with $p=2$.