Some tests for the number of positive zeros and for the numbers of real and complex zeros of a real polynomial

The numbers of positive, real and complex zeros of a polynomial with real coefficients are shown to be related to the number of changes in sign along one or two sequences of determinants whose elements consist of the coefficients of the given polynomial and of its derivative. This in turn is related to the number of negative signs occurring in a certain continued fraction. Some numerical examples illustrate the application of both methods, and the simplification of and the relationship between the two sequences of determinants is also discussed.