Polynomials which are formed by linear combination of the characteristic polynomial of a graph $\,G\,$ and the characteristic polynomials of the vertex-deleted subgraphs of $\,G\,$ have real zeros. The same is true for the linear combination of the matching polynomial of $\,G\,$ and the matching polynomials of the vertex-deleted subgraphs of $\,G\,$. Several statements about the location of the zeros of these polynomials are obtained.