  A novel method with two variations is proposed with which the number of
positive and negative zeros of a polynomial with real coefficients and degree
$n$ can be restricted with significantly better determinacy than that provided
by the Descartes rule of signs and also isolate quite successfully the zeros of
the polynomial. The method relies on solving equations of degree smaller than
that of the given polynomial. One can determine analytically the exact number
of positive and negative zeros of a polynomial of degree up to and including
five and also fully isolate the zeros of the polynomial analytically and with
one of the variations of the method, one can analytically approach polynomials
of degree up to and including nine by solving equations of degree no more than
four. For polynomials of higher degree, either of the two variations of the
method should be applied recursively. Full classification of the roots of the
cubic equation, together with their isolation intervals, is presented. Numerous
examples are given.
