It has been shown recently that there is a new type of codimension one bifurcation, called the singularity-induced bifurcation (SIB) arising in parameter dependent differential-algebraic equations (DAEs) of the form (x) over dot = f and 0 = g, and which occurs generically when an equilibrium path of the DAE crosses the singular surface defined by g = 0 and det g(y) = 0. The SIB refers to a stability change of the DAE owing to some eigenvalue of a related linearization diverging to infinity when the jacobian g(y) is singular. In this article an improved version (Theorem 1.1) of the SIB theorem with its simple proof is given, based on a decomposition theorem (Theorem 2.1) of parameter dependent polynomials.