Let $P(x)=x^3-21x^2+5x+2$. I want to show that $P(x)$ is irreducible over $\mathbb Q$.
Using the Mod $p$ Test, I can reduce $P(x)$ to $\bar P(x)=x^3+2x+2$ in $\mathbb Z_3[x]$, and show that $\bar P(x)$ has no zeroes in $\mathbb Z_3$ by trying $0,1,2$. So $\bar P(x)$ is irreducible over $\mathbb Z_3$, and deg$(\bar P(x))$=deg$(P(x))$, thus by the Mod $p$ Test, $P(x)$ is irreducible over $\mathbb Q$.
Is this proof correct?
Well since this is a degree three polynomial all you have to show is that there is no root in $\mathbb{Q}$. By looking at the coefficients you can show that the only possible roots are $\pm 2, \pm1$. Since these are not roots, the polynomial is irreducible.
EDIT: If the polynomial $\sum\limits_{i=0}^n a_ix^i $ has a rational root $\frac{p}{q}$ $(p,q)=1$, then $p|a_0$ and $q|a_n$.