On the Practicality of the Analytical Solutions for all Third- and Fourth-Degree Algebraic Equations with Real Coefficients

Mauricio Chávez-Pichardo (), Miguel A. Martínez-Cruz (), Alfredo Trejo-Martínez, Ana Beatriz Vega-Cruz and Tanya Arenas-Resendiz
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Mauricio Chávez-Pichardo: TecNM—Tecnológico de Estudios Superiores del Oriente del Estado de México, División de Estudios de Posgrado e Investigación y División de Ingeniería en Energías Renovables, La Paz 56400, Mexico
Miguel A. Martínez-Cruz: Instituto Politécnico Nacional, SEPI-ESIME Zacatenco, Unidad Profesional Adolfo López Mateos, Mexico City 07738, Mexico
Alfredo Trejo-Martínez: TecNM—Tecnológico de Estudios Superiores del Oriente del Estado de México, División de Estudios de Posgrado e Investigación y División de Ingeniería en Energías Renovables, La Paz 56400, Mexico
Ana Beatriz Vega-Cruz: Unidad Académica Profesional Chimalhuacán, Universidad Autónoma del Estado de México, Chimalhuacán 56330, Mexico
Tanya Arenas-Resendiz: Instituto Politécnico Nacional, SEPI-ESIME Zacatenco, Unidad Profesional Adolfo López Mateos, Mexico City 07738, Mexico

Mathematics, 2023, vol. 11, issue 6, 1-34

Abstract: In order to propose a deeper analysis of the general quartic equation with real coefficients, the analytical solutions for all cubic and quartic equations were reviewed here; then, it was found that there can only be one form of the resolvent cubic that satisfies the following two conditions at the same time: (1) Its discriminant is identical to the discriminant of the general quartic equation. (2) It has at least one positive real root whenever the general quartic equation is non-biquadratic. This unique special form of the resolvent cubic is defined here as the “Standard Form of the Resolvent Cubic”, which becomes relevant since it allows us to reveal the relationship between the nature of the roots of the general quartic equation and the nature of the roots of all the forms of the resolvent cubic. Finally, this new analysis is the basis for designing and programming efficient algorithms that analytically solve all algebraic equations of the fourth and lower degree with real coefficients, always avoiding the application of complex arithmetic operations, even when these equations have non-real complex roots.

Keywords: quadratic formula; Tartaglia–Cardano Formulae; Ferrari method; the Standard Form of the Resolvent Cubic; polynomials (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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