The Siebeck-Marden-Northshield Theorem and the Real Roots of the Symbolic Cubic Equation

TL;DR

This paper explores how the Siebeck-Marden-Northshield triangle helps determine the real roots of a cubic polynomial by relating geometric properties to the polynomial's coefficients.

Contribution

It introduces a geometric approach using the Siebeck-Marden-Northshield triangle to find the isolation intervals of real roots of cubic polynomials.

Findings

Isolation intervals are explicitly determined by the triangle's properties.

The method links geometric configurations to algebraic root locations.

Provides a new geometric perspective on solving cubic equations.

Abstract

The isolation intervals of the real roots of the symbolic monic cubic polynomial $x^3 + a x^2 + b x + c$ are determined, in terms of the coefficients of the polynomial, by solving the Siebeck-Marden-Northshield triangle - the equilateral triangle that projects onto the three real roots of the cubic polynomial and whose inscribed circle projects onto an interval with endpoints equal to stationary points of the polynomial