I guess that if a multivariate polynomial is irreducible, then any specialization (substituting a scalar for one or more of the variables) must be irreducible, since I don't know how else to explain (paraphrased from pg. 15, Algebraic Curves and Riemann Surfaces by Rick Miranda):
a nonsingular homogeneous polynomial $F(x,y,z)$ is automatically irreducible . . . then each of $F(1,y,z),F(x,1,z),F(x,y,1)$ define smooth irreducible affine plane curves.
However, I can't help but feel that some specializations could "accidentally" be reducible. Is there a nice way to see that all of the specializations must be irreducible?