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Master's Dissertation
DOI
https://doi.org/10.11606/D.55.2018.tde-02022018-092819
Document
Author
Full name
Livia Novaes Teixeira Passos
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2017
Supervisor
Committee
Artioli, Vanessa Rolnik (President)
Bonotto, Everaldo de Mello
Gonçalves, Jean Piton
Nascimento, Érica Regina Filletti
Title in Portuguese
Soluções analíticas e numéricas de equações polinomiais
Keywords in Portuguese
Equações algébricas
Equações polinomiais
Fórmulas resolventes
Newton Bairstow
Soluções numéricas
Abstract in Portuguese
As equações polinomiais são estudadas desde a antiguidade e atualmente são utilizadas, por exemplo, para modelar problemas do cotidiano nas mais variadas áreas do conhecimento. As técnicas de solução de equações polinomiais nem sempre são triviais, principalmente quando envolvem equações de alta ordem e raízes complexas. O ensino desse tema no Ensino Básico é limitado a equações de segundo ou terceiro grau e coeficientes inteiros, o que restringe a aplicação em problemas mais realistas. Assim, o objetivo deste trabalho é trazer uma contribuição aos estudantes, aos professores do Ensino Básico e aos demais interessados, apresentando um material que aborde técnicas de resolução para equação polinomial de diversas naturezas. Iniciamos por uma revisão dos números complexos e dos polinômios, suas operações e propriedades. Embasamos o trabalho com teoremas e permeamos de exemplos com um crescente grau de dificuldade. Dividimos as técnicas de resolução em analíticas e numéricas. Entre as primeiras, tratamos das relações de Girard, das fórmulas resolventes e de alguns casos particulares de equações. Entre as técnicas numéricas, estudamos o método de Newton, o método das secantes e o método de Newton-Bairstow, este último para encontrar raízes complexas.
Title in English
Analytical and numerical solutions of polynomial equations
Keywords in English
Algebraic equations
Newton Bairstow
Numerical solutions
Polynomial equations
Solving formulas
Abstract in English
Polynomial equations have been studied since antiquity and are currently used, for example, to model everyday problems in the most varied areas of knowledge. The solution techniques of polynomial equations are not always trivial, especially when they involve high order equations and complex roots. The teaching of this subject in Basic Education is limited to second or third degree equations and integer coefficients, which restricts the application to more realistic problems. Thus, the objective of this work is to bring a contribution to students, teachers of Basic Education and other interested parties, presenting a material that treats of resolution techniques for polynomial equation of different natures. We begin with a review of complex numbers and polynomials, their operations and properties. We support the work with theorems and permeate examples with an increasing degree of difficulty. We divide the techniques of resolution into analytical and numerical. Among the first, we deal with Girards relations, the resolvent formulas, and some particular cases of equations. Among numerical techniques, we studied the Newton method, the secant method, and the Newton-Bairstow method, the last one to find complex roots.
 
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Publishing Date
2018-02-02
 
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