In this work, we prove a refinement of the Gallai-Edmonds structure theorem
for weighted matching polynomials by Ku and Wong. Our proof uses a connection
between matching polynomials and branched continued fractions. We also show how
this is related to a modification by Sylvester of the classical Sturm's theorem
on the number of zeros of a real polynomial in an interval. In addition, we
obtain some other results about zeros of matching polynomials