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 |
Fourth Degree Polynomial, Nearly Simultaneous Calculations, New Four Solutions, New Theorems, Solving Quartic Equation
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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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TY - JOUR T1 - New Theorems and Formulas to Solve Fourth Degree Polynomial Equation in General Forms by Calculating the Four Roots Nearly Simultaneously AU - Yassine Larbaoui Y1 - 2023/11/17 PY - 2023 N1 - https://doi.org/10.11648/j.ajam.20231106.11 DO - 10.11648/j.ajam.20231106.11 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 95 EP - 105 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20231106.11 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. VL - 11 IS - 6 ER -