A Complete Review of the General Quartic Equation with Real Coefficients and Multiple Roots

Mauricio Chávez-Pichardo, Miguel A. Martínez-Cruz, Alfredo Trejo-Martínez, Daniel Martínez-Carbajal 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 C.P. 56400, Mexico
Miguel A. Martínez-Cruz: Instituto Politécnico Nacional, SEPI-ESIME Zacatenco, Unidad Profesional Adolfo López Mateos, Mexico City C.P. 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 C.P. 56400, Mexico
Daniel Martínez-Carbajal: 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 C.P. 56400, Mexico
Tanya Arenas-Resendiz: Instituto de Estudios Superiores de la Ciudad de México Rosario Castellanos IESRC, Mexico City C.P. 02440, Mexico

Mathematics, 2022, vol. 10, issue 14, 1-24

Abstract: This paper presents a general analysis of all the quartic equations with real coefficients and multiple roots; this analysis revealed some unknown formulae to solve each kind of these equations and some precisions about the relation between these ones and the Resolvent Cubic; for example, it is well-known that any quartic equation has multiple roots whenever its Resolvent Cubic also has multiple roots; however, this analysis reveals that any non-biquadratic quartic equation and its Resolvent Cubic always have the same number of multiple roots; additionally, the four roots of any quartic equation with multiple roots are real whenever some specific forms of its Resolvent Cubic have three non-negative real roots. This analysis also proves that any method to solve third-degree equations is unnecessary to solve quartic equations with multiple roots, despite the existence of the Resolvent Cubic; finally, here is developed a generalized variation of the Ferrari Method and the Descartes Method, which help to avoid complex arithmetic operations during the resolution of any quartic equation with real coefficients, even though this equation has non-real roots; and a new, more simplified form of the discriminant of the quartic equations is also featured here.

Keywords: quartic equations; biquadratic equations; Resolvent Cubic; multiple roots; algebraic equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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