TITLE: How to prove that $3x^5-12x^4+21x^3-9x+2$ is irreducible over the rationals?
QUESTION [5 upvotes]: Prove that the polynomial $3x^5-12x^4+21x^3-9x+2$ is irreducible over the rationals
$p=2: f(x)=x^5+x^3+x$..this has root..So this is not irreducible over $Z_2$
so Can I conclude $f(x)$ is irreducible over the rationals?

REPLY [13 votes]: Try Eisenstein's criterion on $x^5 P(1/x)$.