A simple proof of the Routh test

An elementary proof of the classic Routh method for counting the number of left half-plane and right half-plane zeros of a real coefficient polynomial P/sub n/(s) of degree n is given. Such a proof refers to the polynomials P/sub i/(s) of degree i/spl les/n formed from the entries of the rows of order i and i-1 of the relevant Routh array. In particular, it is based on the consideration of an auxiliary polynomial P/sub i/(s; q), linearly dependent on a real parameter q, which reduces to either polynomial P/sub i/(s) or to polynomial P/sub i-1/(s) for particular values of q. In this way, it is easy to show that i-1 zeroes of P/sub i/(s) lie in the same half-plane as the zeros of P/sub i/(s), and the remaining zero lies in the left or in the right half-plane according to the sign of the ratio of the leading coefficients of P/sub i/(s) and P/sub i-1/(s). By successively applying this property to all pairs of polynomials in the sequence, starting from P/sub o/(s) and P/sub 1/(s), the standard rule for determining the zero distribution of P/sub n/(s) is immediately derived.