On finding complex roots of polynomials using the Routh array

We present a method, based on the Routh stability criterion, for transforming the task of finding the complex roots of a polynomial of even degree n with real coefficients to that of: (a) finding their real parts as the n/2 real roots of a polynomial of degree n(n−1)/2; (b) for each real part, calculating the corresponding imaginary part as a square root. We illustrate the method for polynomials of degree 4 and indicate its application to polynomials of any even degree.