The negative Routh test and its application to the cases of vanishing leading elements and imaginary roots

The ordinary Routh test furnishes in a straightforward manner the number r_{+} of positive roots of a polynomial of degree n . A modification that allows one to obtain the number r_{-} of negative roots in the same way is presented. Thus, the number r o of (purely) imaginary roots is immediately found as n-(r_{+}+r_{-}) . The modified test is especially useful when there is a first column vanishing element.