Isolation Intervals of the Real Roots of the Parametric Cubic Equation and Improved Complete Root Classification

TL;DR

This paper presents a method to find exact isolation intervals and classify the real roots of cubic equations using simple functions of coefficients, avoiding numerical approximations and discriminant analysis.

Contribution

It introduces a new approach to determine root intervals and signs of roots for cubic polynomials based on auxiliary quadratic equations and coefficient conditions.

Findings

Isolation intervals are expressed through simple functions of coefficients.

A complete root classification with sign and interval information is provided.

The method is illustrated with examples, including Rayleigh wave equations.

Abstract

The isolation intervals of the real roots of the real symbolic monic cubic polynomial $p(x) = x^3 + a x^2 + b x + c\,\,$ are found in terms of simple functions of the coefficients of the polynomial (such as: $-a$, $-a/3$, $-c/b$, $\pm \sqrt{-b}$, when $b$ is negative), and the roots of some auxiliary quadratic equations whose coefficients are also simple functions of the coefficients of the cubic. All possible cases are presented with clear and very detailed diagrams. It is very easy to identify which of these diagrams is the relevant one for any given cubic equation and to read from it the isolation intervals of the real roots of the equation. A much-improved complete root classification, addressing the signs (together with giving the isolation intervals) of the individual roots, is also presented. No numerical approximations or root finding techniques are used. Instead of considering the discriminant of the cubic, criterion for the existence of a single real root or three real roots is found as conditions on the coefficients of the cubic, resulting from the roots of the auxiliary quadratic equations. It is also shown that, if a cubic equation has three real roots, then these lie in an interval $I$ such that $\sqrt{3}\sqrt{a^2/3 - b} \le I \le 2 \sqrt{a^2/3 - b}$, independent of $c$. A detailed algorithm for applying the method for isolation of the roots of the cubic is also given and it is illustrated through examples, including the full mathematical analysis of the cubic equation associated with the Rayleigh elastic waves and finding the isolation intervals of its real roots.