A polynomial is said to be of type (p1, p2, p3) relative to the unit circle if it has p1 zeros interior to, p2 on, and p3 exterior to the unit circle. Stability criteria frequently arise where a polynomial or a family of polynomials must be shown to be of type (p1, p2, 0) or of type (p1, 0, 0). Here we reconsider the practical problem of showing that a polynomial is of one or other of these types, and we show that the testing of a polynomial of degree n may always be reduced to the testing of one of degree n−1. The simplicity of the method is illustrated by its application to several well known difference schemes for partial differential equations.