  The free singularity locus of a noncommutative polynomial f is defined to be
the sequence $Z_n(f)=\{X\in M_n^g : \det f(X)=0\}$ of hypersurfaces. The main
theorem of this article shows that f is irreducible if and only if $Z_n(f)$ is
eventually irreducible. A key step in the proof is an irreducibility result for
linear pencils. Apart from its consequences to factorization in a free algebra,
the paper also discusses its applications to invariant subspaces in
perturbation theory and linear matrix inequalities in real algebraic geometry.
