On the Zeros of Polynomials and the Degree of Stability of Linear Systems

Abstract

A method is presented for finding the zeros of any nth degree real or complex polynomial as its coefficients (or their parameters) are varied. Several new theorems are presented. The method is based on new theorems, some concerning determinants, and on known theorems. It is applied to characteristic equations of linear physical systems with or without feedback. Root trajectories in the complex plane can be visualized without plotting them from ``root trajectory diagrams'' which show curve families for real or complex roots of a polynomial equation in a plane having two coefficients of the polynomial equation as coordinates. The diagrams can be used in the converse manner to determine the coefficient values corresponding to a desired dominant complex pair or real root.