The General Polynomial Equation

Galois Theory ◽  
2015 ◽  
pp. 227-242
Keyword(s):  
Polynomial Equation ◽  
General Polynomial

Unified Functional Method for Solving General Polynomial Equations of Degree Less Than Five

2021 ◽  
Vol 6 (2) ◽  
pp. 924
Author(s):  
Dozie Felix Nwosu ◽  
Odilichukwu Christian Okoli ◽  
Amaka Monica Ezeonyebuchi ◽  
Ababu Teklemariam Tiruneh
Keyword(s):  
Polynomial Equation ◽  
Functional Method ◽  
Auxiliary Function ◽  
Parameter Method ◽  
Computational Formula ◽  
Polynomial Equations ◽  
Standard Formula ◽  
General Polynomial ◽  
Unified Method

A unified method for solving that incorporate a computational formula that relate the coefficients of the depressed equation and the coefficients of the standard polynomial equation is proposed in this study. This is to ensure that this method is valid for all   It shall apply the undetermined parameter method of auxiliary function to obtain solutions to these polynomial equations of degree less than five in one variable.  In particular, the result of our work is a unification and improvement on the work of several authors in the sense that only applicable for the case of polynomial equation of degree one. Finally, our results improve and generalize the result by applying standard formula methods for solving higher degree polynomials. It is recommended that the effort should be made toward providing other variant methods that are simpler and friendly.

scholarly journals Finding Zeros of Quartic Polynomials by Using Radicals

2021 ◽  
Author(s):  
Sureyya Sahin
Keyword(s):  
Polynomial Equation ◽  
Lower Degree ◽  
Polynomial Equations ◽  
Single Variable ◽  
Degree Polynomial ◽  
Numerical Example ◽  
General Polynomial ◽  
Quartic Polynomial ◽  
Quartic Polynomials

We present a technique for finding roots of a quartic general polynomial equation of a single variable by using radicals. The solution of quartic polynomial equations requires knowledge of lower degree polynomial equations; therefore, we study solving polynomial equations of degree less than four as well. We present self-reciprocal polynomials as a specialization and additionally solve numerical example.

scholarly journals Finding Zeros of Quartic Polynomials by Using Radicals

2021 ◽  
Author(s):  
Sureyya Sahin
Keyword(s):  
Polynomial Equation ◽  
Lower Degree ◽  
Polynomial Equations ◽  
Single Variable ◽  
Degree Polynomial ◽  
Numerical Example ◽  
General Polynomial ◽  
Quartic Polynomial ◽  
Quartic Polynomials

We present a technique for finding roots of a quartic general polynomial equation of a single variable by using radicals. The solution of quartic polynomial equations requires knowledge of lower degree polynomial equations; therefore, we study solving polynomial equations of degree less than four as well. We present self-reciprocal polynomials as a specialization and additionally solve numerical example.

Picking genuine zeros of cubics in the Tschirnhaus method

2016 ◽  
Vol 100 (547) ◽  
pp. 48-53
Author(s):  
Raghavendra G. Kulkarni
Keyword(s):  
Polynomial Equation ◽  
German Mathematician ◽  
General Polynomial ◽  
Polynomial Transformation ◽  
Primary Mission

In 1683, the German mathematician Ehrenfried Walther von Tschirnhaus introduced a polynomial transformation which, he claimed, would eliminate all intermediate terms in a polynomial equation of any degree, thereby reducing it to a binomial form from which roots can easily be extracted [1]. As mathematicians at that time were struggling to solve quintic equations in radicals with no sign of any success, the Tschirnhaus transformation gave them some hope, and in 1786, Bring was able to reduce the general quintic to the form x5 + ax + b = 0, even though he didn't succeed in his primary mission of solving it. It seems Bring's work got lost in the archives of University of Lund. Unaware of Bring's work, Jerrard (1859) also arrived at the same form of the quintic using a quartic Tschirnhaus transformation [2]. From the works of Abel (1826) and Galois (1832) it is now clear that the general polynomial equation of degree higher than four cannot be solved in radicals.

Optimisation of the Oil Extraction from Nigella sativa Seeds Using Response Surface Methodology

2017 ◽  
Vol 68 (2) ◽  
pp. 331-336
Author(s):  
Gabriela Isopencu ◽  
Mirela Marfa ◽  
Iuliana Jipa ◽  
Marta Stroescu ◽  
Anicuta Stoica Guzun ◽  
...  
Keyword(s):  
Response Surface Methodology ◽  
Response Surface ◽  
Nigella Sativa ◽  
Polynomial Equation ◽  
Herbaceous Plant ◽  
Box Behnken Design ◽  
Pareto Analysis ◽  
Black Cumin ◽  
Mediterranean Countries ◽  
Experimental Values

Nigella sativa, also known as black cumin, an annual herbaceous plant growing especially in Mediterranean countries, has recently gained considerable interest not only for its use as spice and condiment but also for its healthy properties of the fixed and essential oil and its potential as a biofuel. Nigella sativa seeds fixed oil, due to its high content in linoleic acid followed by oleic and palmitic acid, could be beneficial to human health. The objective of this study is to determine the optimum conditions for the solvent extraction of Nigella sativa seeds fixed oil using a three-level, three-factor Box-Behnken design (BBD) under response surface methodology (RSM). The obtained experimental data, fitted by a second-order polynomial equation were analysed by Pareto analysis of variance (ANOVA). From a total of 10 coefficients of the statistical model only 5 are important. The obtained experimental values agreed with the predicted ones.

Viscosity of Dimethyl Sulfoxide + 1,4 Dimethylbenzene System

2008 ◽  
Vol 59 (1) ◽  
pp. 45-48
Author(s):  
Oana Ciocirlan ◽  
Olga Iulian
Keyword(s):  
Free Energy ◽  
Dimethyl Sulfoxide ◽  
Atmospheric Pressure ◽  
Entire Range ◽  
Polynomial Equation ◽  
Energy Of Activation ◽  
Excess Gibbs Free Energy ◽  
Experimental Measurements ◽  
Gibbs Energy Of Activation ◽  
Free Energy Of Activation

This paper reports the viscosities measurements for the binary system dimethyl sulfoxide + 1,4-dimethylbenzene over the entire range of mole fraction at 298.15, 303.15, 313.15 and 323.15 K and atmospheric pressure. The experimental viscosities were correlated with the equations of Grunberg-Nissan, Katti-Chaudhri, Hind, Soliman and McAllister; the adjustable binary parameters have been obtained. The excess Gibbs energy of activation of viscous flow (G*E) has been calculated from the experimental measurements and the results were fitted to Redlich-Kister polynomial equation. The obtained negative excess Gibbs free energy of activation and negative Grunberg-Nissan interaction parameter are discussed in structural and interactional terms.

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