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 - Resolvent Cubic
KW - algebraic equations
KW - biquadratic equations
KW - multiple roots
KW - quartic equations
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
SN - 2227-7390
VL - 10
JO - Mathematics
JF - Mathematics
IS - 14
M1 - 2377
ER -