An algorithm for generating a sequence of polynomials of descending degree is illustrated. This algorithm becomes the usual Euclid algorithm for particular parameter values. A split form of the basic recursion is derived, and a stability test is presented in table form. A geometrical proof of the rules for finding the right half-plane and left half-plane zeros of the original polynomial is given. The relationship with the Routh algorithm is pointed out. The possible use of the suggested algorithm for flexible model reduction is also outlined.