I'm just now trying to do another related exam question:

Show that g = X^3 + 2X - 1 is irreducible over rationals.

If x = b/c, b must be a factor of -1, c must be a factor of 1.

Possible factors of -1: +1, -1

Possible factors of 1: +1, -1

List of possible rational roots:

Direct checking shows that none of these are a root and hence g has no rational root.

Therefore since there are no rational roots g cannot be irreducible over the rationals.

Is this the right approach for this question?

Thanks in advance!