TY - JOUR
T1 - On the Practicality of the Analytical Solutions for all Third- and Fourth-Degree Algebraic Equations with Real Coefficients
AU - Chávez-Pichardo, Mauricio
AU - Martínez-Cruz, Miguel A.
AU - Trejo-Martínez, Alfredo
AU - Vega-Cruz, Ana Beatriz
AU - Arenas-Resendiz, Tanya
N1 - Publisher Copyright:
© 2023 by the authors.
PY - 2023/3
Y1 - 2023/3
N2 - 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.
AB - 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.
KW - Ferrari method
KW - Tartaglia–Cardano Formulae
KW - polynomials
KW - quadratic formula
KW - the Standard Form of the Resolvent Cubic
UR - http://www.scopus.com/inward/record.url?scp=85151506657&partnerID=8YFLogxK
U2 - 10.3390/math11061447
DO - 10.3390/math11061447
M3 - Artículo
AN - SCOPUS:85151506657
SN - 2227-7390
VL - 11
JO - Mathematics
JF - Mathematics
IS - 6
M1 - 1447
ER -