Our work in Chapters 1 and 2 revealed some similarities between the properties of the matchings and characteristic polynomials of a graph. In particular these two polynomials are the same for forests, and satisfy some similar recursions. The main result of this chapter, Theorem 1.1, provides good reason for this—it implies that the matchings polynomial of any connected graph is a divisor of the matchings polynomial of a tree. Hence every matchings polynomial divides the characteristic polynomial of some graph. It follows at once that the zeros of μ(G, x) are real.