  Godsil observed the simple fact that the multiplicity of 0 as a root of the
matching polynomial of a graph coincides with the classical notion of
deficiency. From this fact he asked to what extent classical results in
matching theory generalize, replacing "deficiency" with multiplicity of
$\theta$ as a root of the matching polynomial. We prove an analogue of the
Stability Lemma for any given root, which describes how the matching structure
of a graph changes upon deletion of a single vertex. An analogue of Gallai's
Lemma follows. Together these two results imply an analogue of the
Gallai-Edmonds Structure Theorem. Consequently, the matching polynomial of a
vertex transitive graph has simple roots.
