On the Cubic Equation with its Siebeck--Marden--Northshield Triangle and the Quartic Equation with its Tetrahedron

TL;DR

This paper explores the geometric relationships of roots of cubic and quartic polynomials using their associated regular geometric figures, providing new root classification, bounds, and localization methods based on geometric interpretations.

Contribution

It introduces a geometric approach to root analysis of cubics and quartics, including root classification, bounds, and localization techniques using associated triangles and tetrahedra.

Findings

Established Viète trigonometric formulas for cubic roots via geometric rotation.

Provided a detailed classification of quartic roots based on coefficients.

Derived bounds and localization intervals for quartic roots in terms of coefficients.

Abstract

The real roots of the cubic and quartic polynomials are studied geometrically with the help of their respective Siebeck--Marden--Northshield equilateral triangle and regular tetrahedron. The Vi\`ete trigonometric formulae for the roots of the cubic are established through the rotation of the triangle by variation of the free term of the cubic. A very detailed complete root classification for the quartic $x^4 + ax^3 + bx^2 + cx + d$ is proposed for which the conditions are imposed on the individual coefficients $a$, $b$, $c$, and $d$. The maximum and minimum lengths of the interval containing the four real roots of the quartic are determined in terms of $a$ and $b$. The upper and lower root bounds for a quartic with four real roots are also found: no root can lie farther than $(\sqrt{3}/4)\sqrt{3a^2 - 8b}\, $ from $-a/4$. The real roots of the quartic are localized by finding intervals containing at most two roots. The end-points of these intervals depend on $a$ and $b$ and are roots of quadratic equations -- which makes this localization helpful for quartic equations with complicated parametric coefficients.