Irreducible polynomial of degree 3

Question

$$P(X)= 21X^3 -3X^2+2X+9$$ 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 $Q[X]$.

Answer 1

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.