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.

THE KINEMATIC SYNTHESIS OF STEERING MECHANISMS

2000 ◽  
Vol 24 (3-4) ◽  
pp. 453-476 ◽  
Author(s):  
Jin Yao ◽  
Jorge Angeles
Keyword(s):  
Polynomial Equation ◽  
Global Optimum ◽  
Polynomial Equations ◽  
Kinematic Synthesis ◽  
Local Optima ◽  
Speed Reduction ◽  
Design Variables ◽  
Transmission Quality ◽  
Four Bar Linkage ◽  
Maximum Transmission

We propose a computational-kinematics approach based on elimination procedures to synthesize a steering four-bar linkage. In this regard, we aim at minimizing the root-mean square error of the synthesized linkage in meeting the steering condition over a number of linkage configurations within the linkage range of motion. A minimization problem is thus formulated, whose normality conditions lead to two polynomial equations in two unknown design variables. Upon eliminating one of these two variables, a monovariate polynomial equation is obtained, whose roots yield all locally-optimum linkages. From these roots, the global optimum, as well as unfeasible local optima, are readily identified. The global optimum, however, turns out to be impractical because of the large differences in its link lengths, which we refer to as dimensional unbalance. To cope with this drawback, we use a kinematically-equivalent focal mechanism, i.e., a six-bar linkage with an input-output function identical to that of the four-bar linkage. Given that the synthesized linkage requires a rotational input, as opposed to most existing steering linkages, which require a translational input, we propose a spherical four-bar linkage to drive the steering linkage. The spherical linkage is synthesized so as to yield a speed reduction as close as possible to 2:1 and to have a maximum transmission quality.

scholarly journals Galois theory for general systems of polynomial equations

2019 ◽  
Vol 155 (2) ◽  
pp. 229-245 ◽  
Author(s):  
A. Esterov
Keyword(s):  
Galois Theory ◽  
General System ◽  
Mixed Volume ◽  
Polynomial Equations ◽  
Systems Of Equations ◽  
Change Of Variables ◽  
General Polynomial ◽  
General Systems ◽  
Systems Of Polynomial Equations

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel–Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive. The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.

scholarly journals Accurate Computation of Periodic Regions' Centers in the General M-Set with Integer Index Number

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Wang Xingyuan ◽  
He Yijie ◽  
Sun Yuanyuan
Keyword(s):  
Polynomial Equation ◽  
Negative Integer ◽  
The Other ◽  
Index Number ◽  
Polynomial Equations ◽  
Decimal Point ◽  
Numerical Accuracy ◽  
Index Numbers ◽  
Accurate Computation ◽  
The Relationship

This paper presents two methods for accurately computing the periodic regions' centers. One method fits for the general M-sets with integer index number, the other fits for the general M-sets with negative integer index number. Both methods improve the precision of computation by transforming the polynomial equations which determine the periodic regions' centers. We primarily discuss the general M-sets with negative integer index, and analyze the relationship between the number of periodic regions' centers on the principal symmetric axis and in the principal symmetric interior. We can get the centers' coordinates with at least 48 significant digits after the decimal point in both real and imaginary parts by applying the Newton's method to the transformed polynomial equation which determine the periodic regions' centers. In this paper, we list some centers' coordinates of general M-sets'k-periodic regions(k=3,4,5,6)for the index numbersα=−25,−24,…,−1, all of which have highly numerical accuracy.

Forward displacement analysis of a new 1CCC–5SPS parallel mechanism using Gröbner theory

2009 ◽  
Vol 223 (5) ◽  
pp. 1233-1241 ◽  
Author(s):  
D Gan ◽  
Q Liao ◽  
J S Dai ◽  
S Wei ◽  
L D Seneviratne
Keyword(s):  
Parallel Mechanism ◽  
Polynomial Equation ◽  
Polynomial Equations ◽  
Geometrical Constraint ◽  
Degree Polynomial ◽  
Forward Displacement ◽  
Constraint Equations ◽  
Moving Platform ◽  
Forward Kinematic ◽  
Forward Displacement Analysis

A new parallel mechanism 1CCC–5SPS which has distance and angle constraints is introduced in this article. Degree of freedom and forward kinematic analysis of this new parallel mechanism are presented, in which four equivalent polynomial equations are obtained from the original six geometrical constraint equations. The Gröbner basis theory is used with the four equations and the problem of forward displacement is reduced to a 40th degree polynomial equation in a single unknown from a constructed 10 × 10 Sylvester's matrix which is small in size, from which 40 different locations of the moving platform can be derived. A numerical example confirms the efficiency of the procedure.

Direct Kinematic Singularities of 3-3 Stewart-Gough Platforms

2005 ◽  
Vol 219 (4) ◽  
pp. 311-324 ◽  
Author(s):  
C H Liu ◽  
J Chiu
Keyword(s):  
Parallel Manipulator ◽  
Degrees Of Freedom ◽  
Polynomial Equation ◽  
Numerical Results ◽  
Polynomial Equations ◽  
Kinematic Singularity ◽  
Cubic Polynomial ◽  
Singular Configuration ◽  
Kinematic Singularities ◽  
Stewart Gough Platform

In this article, a method to locate direct kinematic singularities of a 3-3 Stewart-Gough parallel manipulator (called a Stewart manipulator henceforth) is proposed. The Stewart manipulator is first replaced by an analogous manipulator, the 3PRPS parallel manipulator, and as the first three active joints of this manipulator remain fixed, this manipulator reduces to an asymmetric 3RPS parallel manipulator. With all moving platform's degrees of freedom, except its height, properly specified, there exists at least one height that gives rise to direct kinematic singularity of the asymmetric 3RPS manipulator and this height is a root of a cubic polynomial equation. The procedure to locate direct kinematic singularities thus reduces to solving cubic polynomial equations. Numerical results show that every singular configuration of the asymmetric 3RPS manipulator thus-determined is also a singular configuration of the 3-3 Stewart-Gough platform.

scholarly journals A combined methodology for the approximate estimation of the roots of the general sextic polynomial equation

2021 ◽  
Author(s):  
Giorgos P. Kouropoulos
Keyword(s):  
Mathematical Model ◽  
Regression Analysis ◽  
Polynomial Equation ◽  
Approximate Solutions ◽  
The Other ◽  
Polynomial Equations ◽  
Polynomial Coefficients ◽  
Approximate Estimation ◽  
Predetermined Interval

Abstract In this article we present the methodology, according to which it is possible to derive approximate solutions for the roots of the general sextic polynomial equation as well as some other forms of sextic polynomial equations that normally cannot be solved by radicals; the approximate roots can be expressed in terms of polynomial coefficients. This methodology is a combination of two methods. The first part of the procedure pertains to the reduction of a general sextic equation H(x) to a depressed equation G(y), followed by the determination of solutions by radicals of G(y) which does not include a quintic term, provided that the fixed term of the equation depends on its other coefficients. The second method is a continuation of the first and pertains to the numerical correlation of the roots and the fixed term of a given sextic polynomial P(x) with the radicals and the fixed term of the sextic polynomial Q(x), where the two polynomials P(x) and Q(x) have the same coefficients except for the fixed term which might be different. From the application of the methodology presented above, the following formulation is derived; For any given general sextic polynomial equation P with coefficients within the interval [a, b], a defined polynomial equation Q corresponds which has equal coefficients to P except for its fixed term which might be different and dependent on the other coefficients so that Q has radical solutions. If we assume a pair of equations P, Q with coefficients within a predetermined interval [a, b], the numerical correlation through regression analysis of the radicals of Q, the roots of P and the fixed terms of P, Q, leads to the derivation of a mathematical model for the approximate estimation of the roots of sextic equations whose coefficients belong to the interval [a, b].

scholarly journals Exact solutions of sixth and fifth degree equations

2021 ◽  
Author(s):  
Mohammed El Amine MONIR
Keyword(s):  
Exact Solutions ◽  
Algebraic Polynomial ◽  
Polynomial Equation ◽  
Polynomial Equations ◽  
Degree Polynomial ◽  
The Real ◽  
Absolute Method ◽  
Quintic Equation ◽  
Degree Equation

Abstract The real problematic with algebraic polynomial equations is how to exactly solve any sixth and fifth degree polynomial equations. In this study, we give a new absolute method that presents a new decomposition to exactly solve a sixth degree polynomial equation, while the corresponding fifth degree equation can be easily transformed into a sixth degree equation of this kind (sixth degree equation solvable by this method), then the sextic equation (sixth degree equation) obtained will be solved by applying the principles of this method; moreover, the solutions of the quintic equation (fifth degree equation) will be easily deduced.

scholarly journals Praxéologies du professeur : analyse comparative de manuels scolaires dans l'enseignement des équations polynomiales du premier degréTeacher's Praxeologies: Comparative Analysis of the Textbook in Teaching First-Degree Polynomial Equations

2020 ◽  
Vol 22 (4) ◽  
pp. 224-231
Author(s):  
Edelweis Jose Tavares Barbosa ◽  
Anna Paula de Avelar Brito Lima
Keyword(s):  
Polynomial Equation ◽  
Classroom Teachers ◽  
Polynomial Equations ◽  
Degree Polynomial ◽  
Mots Clés ◽  
Anthropological Theory ◽  
The Common ◽  
Analyse Comparative ◽  
Théorie Anthropologique Du Didactique ◽  
Anthropological Theory Of Didactics

RésuméLe but de cet article est d'analyser, de manière comparative, les livres didactiques et les praxéologies mises en place par les enseignants dans leur pratique pédagogique, concernant l'enseignement des équations polynomiales du premier degré. Cette étude est faite dans le cadre de la théorie anthropologique du didactique (TAD) proposée par Yves Chevallard et ses collaborateurs (1999, 2002, 2009, 2010). La méthodologie est basée sur une approche ethnographique qualitative, dans laquelle les organisations mathématiques et didactiques de trois enseignants sont analysées en les comparant à celles des livres de référence. Les résultats indiquent qu'il existe une certaine conformité entre les praxéologies à enseigner, proposées par les auteurs des manuels scolaires et les praxéologies effectivement enseignées par les professeurs en classe. Les enseignants sont les organisateurs des tâches, des techniques et de la technologie de complexité croissante (FONSECA, 2004) qui sont rendus routinières ou problématiques en classe. La résolution d’une équation polynomiale du premier degré du type ax+b=c a été le point commun des trois professeurs, bien que deux des trois enseignants aient aussi travaillé des équations du type a1x+b1=a2x+b2.Mots-clés : Livres didactiques, Équations polynomiales du premier degré, Théorie Anthropologique du didactique.AbstractThe aim of this article was to analyze, comparatively, praxeologies in didactic books and praxeologies carried out by the teacher, concerning the teaching of polynomial equations of the first degree. This study is done within the framework of the Anthropological Theory of Didactics (ATD), proposed by Yves Chevallard and his collaborators (1999, 2002, 2009, 2010). The methodology consists of a qualitative ethnographic approach, in which the mathematical and didactic organizations of three teachers were compared with those of their reference books. The results indicate that there is some conformity between the praxeologies to be taught, proposed by the authors of the textbooks, and the praxeologies effectively taught by the teachers in the classroom. Teachers are the organizers of tasks, techniques, and technology of increasing complexity (FONSECA, 2004) that are made routine or problematic in the classroom. The resolution of a first-degree polynomial equation of the type ax+b=c was the common point among the three teachers, although two of the three teachers also worked on equations of the type a1x+b1=a2x+b2.Keywords: Didactic books, Polynomial equations of the first degree, Anthropological theory of didactics.

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