Q:

Irreducible polynomial of degree 3

$$P(X)= 21X^3 -3X^2+2X+9$$
To check whether it is irreducible or not in $Q[X]$.  
Since it's degree $3$ if it has a rational root then it is reducible as one of them would be linear factor; but how to show whether a polynomial of degree three has root or not in $Q[X]$.

A:

After using the rational root theorem to make a shortish (should be less than $12$) list of potential roots, try computing $P(r)$ for each of the potential roots! If none of them are $0$, then the polynomial is irreducible.

