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

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.