# On the Determination of the Number of Positive and Negative Polynomial Zeros and Their Isolation

## TL;DR

This paper introduces two new methods to accurately determine and isolate the positive and negative zeros of polynomials with real coefficients, improving upon traditional rules and enabling analytical solutions up to degree nine.

## Contribution

The paper presents novel variations of a method that significantly enhance the ability to determine and isolate polynomial zeros analytically, surpassing the capabilities of Descartes' rule of signs.

## Key findings

- Exact determination of positive and negative zeros for degree up to five.
- Analytical approach for polynomials up to degree nine.
- Recursive application for higher degree polynomials.

## Abstract

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.