### American Journal of Applied Mathematics

Research Article | | Peer-Reviewed |

### New Theorems and Formulas to Solve Fourth Degree Polynomial Equation in General Forms by Calculating the Four Roots Nearly Simultaneously

Received: Oct. 06, 2023    Accepted: Nov. 02, 2023    Published: Nov. 17, 2023

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Abstract

This paper presents new formulary solutions for fourth degree polynomial equations in general forms, where we present four solutions for any fourth-degree equation with real coefficients, and thereby having the possibility to calculate the four roots of any quartic equation nearly simultaneously. In this paper, the used logic to determine the solutions of a fourth-degree polynomial equation enables to deduce if the polynomial accepts complex roots with imaginary parts different from zero and how many of them there are. As a result, we are proposing six new theorems, where two among them are allowing to calculate the four roots nearly simultaneously for any fourth-degree polynomial equation in simple forms and complete forms, whereas the other four theorems are allowing to deduce the number of complex roots with imaginary parts different from zero before conducting further fetching for the values. Furthermore, the proposed formulas in this paper are building the ground to concretize precise solutions for polynomial equations with degrees higher than four while relying on radical expressions. Each proposed theorem in this paper is presented along with a detailed proof in a scaling manner starting from propositions based on precise formulas whereas building on progressive logic of calculation and deduction. Each formulary solution in proposed theorems is based on a distributed group of radical expressions designed to be neutralized when they are multiplied by each other, which allow the elimination of complexity while reducing degrees of terms. All presented theorems are developed according to a specific logic where we engineer the structure of solutions before forwarding calculations to express the precis formulas of roots.

 DOI 10.11648/j.ajam.20231106.11 Published in American Journal of Applied Mathematics ( Volume 11, Issue 6, December 2023 ) Page(s) 95-105 Creative Commons This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. Copyright Copyright © The Author(s), 2024. Published by Science Publishing Group
Keywords

Fourth Degree Polynomial, Nearly Simultaneous Calculations, New Four Solutions, New Theorems, Solving Quartic Equation

References
 [1] Cardano G. Artis Magnae, Sive de Regulis Algebraicis Liber Unus, 1545. English transl.: The Great Art, or The Rules of Algebra. Translated and edited by T. R. Witmer, MIT Press, Cambridge, Mass; 1968. [2] Dumit D S, and Foote R M. Abstract algebra. John Wiley; 2004. p. 606-616. [3] Osler T J. Cardan polynomials and the reduction of radicals. Mathematics Magazine. 2001; 47 (1): 26-32. [4] Euler L. De formis radicum aequationum cuiusque ordinis coniectatio. 1738. English transl.: A conjecture on the forms of the roots of equations, Translated by J. Bell, Cornell University, NY, USA; 2008. [5] Janson S. Roots of polynomials of degrees 3 and 4; 2010. [6] René D. The Geometry of Rene Descartes with a facsimile of the 1st edition. Courier Corporation; 2012. [7] Lagrange J L. Réflexions sur la résolution algébrique des équations, in: ouvres de Lagrange. J. A. Serret ed & Gauthier-Villars; 1869. Vol. 3. p. 205-421. [8] Faucette W M. A geometric interpretation of the solution of the general quartic polynomial. Amer. Math. Monthly; 1996. vol. 103. p. 51-57. [9] Bewersdor J. Algebra fur Einsteiger, Friedr. Vieweg Sohn Verlag; 2004. English transl.: Galois Theory for Beginners. A Historical Perspective. Translated by Kramer D, American Mathematical Society (AMS), Providence, R. I., USA; 2006. [10] Helfgott H, Helfgott M. A modern vision of the work of cardano and ferrari on quartics. Convergence (MAA). [11] Rosen M. Niels hendrik abel and equations of the fiffth degree. American Mathematical Monthly. 1995; 102 (6); 495-505. doi.org/10.2307/2974763. [12] Garling D J. Galois Theory. Cambridge Univ. Press, Mass., USA; 1986. [13] Grillet P, Abstract Algebra, 2nd Edition, Springer, New York, USA; 2007. doi.org/10.1007/978-0-387-71568-1. [14] van der Waerden B L. Algebra. Springer-Verlag, Vol. 1, 3rd ed. Berlin; 1966. English transl.: Algebra. Translated by J. R. Schulenberg J R and Blum F, Springer-Verlag, New York; 1991. [15] Shmakov S L. A universal method of solving quartic equations. International Journal of Pure and Applied Mathematics. 2011; 71 (2): 251-259. [16] Fathi A, Sharifan N. A classic new method to solve quartic equations. Applied and Computational Mathematics. 2013; 2 (2): 24-27. [17] Tehrani F T. Solution to polynomial equations, a new approach. Applied Mathematics. 2020; 11 (2): 53-66. [18] Nahon Y J. Method for solving polynomial equations. Journal of Applied & Computational Mathematics. 2018; 7 (3): 2-12. [19] Larbaoui Y. New Five Roots to Solve Quantic Equation in General Forms by Using Radical Expressions Along With New Theorems. Cornell University Arxiv. 2022: 1-17. arXiv: 2210.07957 [math. GM]. [20] Larbaoui Y. New Six Solutions to Solve Sixth Degree Polynomial Equation in General Forms by Relying on Radical Expressions. Cornell University Arxiv, 2022: 1-22. arXiv: 2211.08395 [math. GM].
• APA Style

Larbaoui, Y. (2023). New Theorems and Formulas to Solve Fourth Degree Polynomial Equation in General Forms by Calculating the Four Roots Nearly Simultaneously. American Journal of Applied Mathematics, 11(6), 95-105. https://doi.org/10.11648/j.ajam.20231106.11

ACS Style

Larbaoui, Y. New Theorems and Formulas to Solve Fourth Degree Polynomial Equation in General Forms by Calculating the Four Roots Nearly Simultaneously. Am. J. Appl. Math. 2023, 11(6), 95-105. doi: 10.11648/j.ajam.20231106.11

AMA Style

Larbaoui Y. New Theorems and Formulas to Solve Fourth Degree Polynomial Equation in General Forms by Calculating the Four Roots Nearly Simultaneously. Am J Appl Math. 2023;11(6):95-105. doi: 10.11648/j.ajam.20231106.11

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title = {New Theorems and Formulas to Solve Fourth Degree Polynomial Equation in General Forms by Calculating the Four Roots Nearly Simultaneously},
journal = {American Journal of Applied Mathematics},
volume = {11},
number = {6},
pages = {95-105},
doi = {10.11648/j.ajam.20231106.11},
url = {https://doi.org/10.11648/j.ajam.20231106.11},
abstract = {This paper presents new formulary solutions for fourth degree polynomial equations in general forms, where we present four solutions for any fourth-degree equation with real coefficients, and thereby having the possibility to calculate the four roots of any quartic equation nearly simultaneously. In this paper, the used logic to determine the solutions of a fourth-degree polynomial equation enables to deduce if the polynomial accepts complex roots with imaginary parts different from zero and how many of them there are. As a result, we are proposing six new theorems, where two among them are allowing to calculate the four roots nearly simultaneously for any fourth-degree polynomial equation in simple forms and complete forms, whereas the other four theorems are allowing to deduce the number of complex roots with imaginary parts different from zero before conducting further fetching for the values. Furthermore, the proposed formulas in this paper are building the ground to concretize precise solutions for polynomial equations with degrees higher than four while relying on radical expressions. Each proposed theorem in this paper is presented along with a detailed proof in a scaling manner starting from propositions based on precise formulas whereas building on progressive logic of calculation and deduction. Each formulary solution in proposed theorems is based on a distributed group of radical expressions designed to be neutralized when they are multiplied by each other, which allow the elimination of complexity while reducing degrees of terms. All presented theorems are developed according to a specific logic where we engineer the structure of solutions before forwarding calculations to express the precis formulas of roots.
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AB  - This paper presents new formulary solutions for fourth degree polynomial equations in general forms, where we present four solutions for any fourth-degree equation with real coefficients, and thereby having the possibility to calculate the four roots of any quartic equation nearly simultaneously. In this paper, the used logic to determine the solutions of a fourth-degree polynomial equation enables to deduce if the polynomial accepts complex roots with imaginary parts different from zero and how many of them there are. As a result, we are proposing six new theorems, where two among them are allowing to calculate the four roots nearly simultaneously for any fourth-degree polynomial equation in simple forms and complete forms, whereas the other four theorems are allowing to deduce the number of complex roots with imaginary parts different from zero before conducting further fetching for the values. Furthermore, the proposed formulas in this paper are building the ground to concretize precise solutions for polynomial equations with degrees higher than four while relying on radical expressions. Each proposed theorem in this paper is presented along with a detailed proof in a scaling manner starting from propositions based on precise formulas whereas building on progressive logic of calculation and deduction. Each formulary solution in proposed theorems is based on a distributed group of radical expressions designed to be neutralized when they are multiplied by each other, which allow the elimination of complexity while reducing degrees of terms. All presented theorems are developed according to a specific logic where we engineer the structure of solutions before forwarding calculations to express the precis formulas of roots.

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Author Information
• Department of Electrical Engineering, University Hassan 1er, Settat, Morocco

• Section