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_n(s) of degree n is given. Such a proof refers to the polynomials P_i(s) of degree i⩽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_i(s; q), linearly dependent on a real parameter q, which reduces to either polynomial P_i(s) or to polynomial P_i-1(s) for particular values of q. In this way, it is easy to show that i-1 zeroes of P_i(s) lie in the same half-plane as the zeros of P_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_i(s) and P_i-1(s). By successively applying this property to all pairs of polynomials in the sequence, starting from P_o(s) and P₁(s), the standard rule for determining the zero distribution of P_n(s) is immediately derived