TY - JOUR

T1 - A Complete Review of the General Quartic Equation with Real Coefficients and Multiple Roots

AU - Chávez-Pichardo, Mauricio

AU - Martínez-Cruz, Miguel A.

AU - Trejo-Martínez, Alfredo

AU - Martínez-Carbajal, Daniel

AU - Arenas-Resendiz, Tanya

N1 - Publisher Copyright:
© 2022 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2022/7/1

Y1 - 2022/7/1

N2 - 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.

AB - 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.

KW - algebraic equations

KW - biquadratic equations

KW - multiple roots

KW - quartic equations

KW - Resolvent Cubic

UR - http://www.scopus.com/inward/record.url?scp=85134049005&partnerID=8YFLogxK

U2 - 10.3390/math10142377

DO - 10.3390/math10142377

M3 - Artículo

AN - SCOPUS:85134049005

VL - 10

JO - Mathematics

JF - Mathematics

SN - 2227-7390

IS - 14

M1 - 2377

ER -