A Simplified Expression for the Solution of Cubic Polynomial Equations
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Symmetry and the Cubic Formula
SYMMETRY AND THE CUBIC FORMULA DAVID CORWIN The purpose of this talk is to derive the cubic formula. But rather than finding the exact formula, I'm going to prove that there is a cubic formula. The way I'm going to do this uses symmetry in a very elegant way, and it foreshadows Galois theory. Note that this material comes almost entirely from the first chapter of http://homepages.warwick.ac.uk/~masda/MA3D5/Galois.pdf. 1. Deriving the Quadratic Formula Before discussing the cubic formula, I would like to consider the quadratic formula. While I'm expecting you know the quadratic formula already, I'd like to treat this case first in order to motivate what we will do with the cubic formula. Suppose we're solving f(x) = x2 + bx + c = 0: We know this factors as f(x) = (x − α)(x − β) where α; β are some complex numbers, and the problem is to find α; β in terms of b; c. To go the other way around, we can multiply out the expression above to get (x − α)(x − β) = x2 − (α + β)x + αβ; which means that b = −(α + β) and c = αβ. Notice that each of these expressions doesn't change when we interchange α and β. This should be the case, since after all, we labeled the two roots α and β arbitrarily. This means that any expression we get for α should equally be an expression for β. That is, one formula should produce two values. We say there is an ambiguity here; it's ambiguous whether the formula gives us α or β. -
Solving Cubic Polynomials
Solving Cubic Polynomials 1.1 The general solution to the quadratic equation There are four steps to finding the zeroes of a quadratic polynomial. 1. First divide by the leading term, making the polynomial monic. a 2. Then, given x2 + a x + a , substitute x = y − 1 to obtain an equation without the linear term. 1 0 2 (This is the \depressed" equation.) 3. Solve then for y as a square root. (Remember to use both signs of the square root.) a 4. Once this is done, recover x using the fact that x = y − 1 . 2 For example, let's solve 2x2 + 7x − 15 = 0: First, we divide both sides by 2 to create an equation with leading term equal to one: 7 15 x2 + x − = 0: 2 2 a 7 Then replace x by x = y − 1 = y − to obtain: 2 4 169 y2 = 16 Solve for y: 13 13 y = or − 4 4 Then, solving back for x, we have 3 x = or − 5: 2 This method is equivalent to \completing the square" and is the steps taken in developing the much- memorized quadratic formula. For example, if the original equation is our \high school quadratic" ax2 + bx + c = 0 then the first step creates the equation b c x2 + x + = 0: a a b We then write x = y − and obtain, after simplifying, 2a b2 − 4ac y2 − = 0 4a2 so that p b2 − 4ac y = ± 2a and so p b b2 − 4ac x = − ± : 2a 2a 1 The solutions to this quadratic depend heavily on the value of b2 − 4ac. -
A Historical Survey of Methods of Solving Cubic Equations Minna Burgess Connor
University of Richmond UR Scholarship Repository Master's Theses Student Research 7-1-1956 A historical survey of methods of solving cubic equations Minna Burgess Connor Follow this and additional works at: http://scholarship.richmond.edu/masters-theses Recommended Citation Connor, Minna Burgess, "A historical survey of methods of solving cubic equations" (1956). Master's Theses. Paper 114. This Thesis is brought to you for free and open access by the Student Research at UR Scholarship Repository. It has been accepted for inclusion in Master's Theses by an authorized administrator of UR Scholarship Repository. For more information, please contact [email protected]. A HISTORICAL SURVEY OF METHODS OF SOLVING CUBIC E<~UATIONS A Thesis Presented' to the Faculty or the Department of Mathematics University of Richmond In Partial Fulfillment ot the Requirements tor the Degree Master of Science by Minna Burgess Connor August 1956 LIBRARY UNIVERStTY OF RICHMOND VIRGlNIA 23173 - . TABLE Olf CONTENTS CHAPTER PAGE OUTLINE OF HISTORY INTRODUCTION' I. THE BABYLONIANS l) II. THE GREEKS 16 III. THE HINDUS 32 IV. THE CHINESE, lAPANESE AND 31 ARABS v. THE RENAISSANCE 47 VI. THE SEVEW.l'EEl'iTH AND S6 EIGHTEENTH CENTURIES VII. THE NINETEENTH AND 70 TWENTIETH C:BNTURIES VIII• CONCLUSION, BIBLIOGRAPHY 76 AND NOTES OUTLINE OF HISTORY OF SOLUTIONS I. The Babylonians (1800 B. c.) Solutions by use ot. :tables II. The Greeks·. cs·oo ·B.c,. - )00 A~D.) Hippocrates of Chios (~440) Hippias ot Elis (•420) (the quadratrix) Archytas (~400) _ .M~naeobmus J ""375) ,{,conic section~) Archimedes (-240) {conioisections) Nicomedea (-180) (the conchoid) Diophantus ot Alexander (75) (right-angled tr~angle) Pappus (300) · III. -
Analytical Formula for the Roots of the General Complex Cubic Polynomial Ibrahim Baydoun
Analytical formula for the roots of the general complex cubic polynomial Ibrahim Baydoun To cite this version: Ibrahim Baydoun. Analytical formula for the roots of the general complex cubic polynomial. 2018. hal-01237234v2 HAL Id: hal-01237234 https://hal.archives-ouvertes.fr/hal-01237234v2 Preprint submitted on 17 Jan 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Analytical formula for the roots of the general complex cubic polynomial Ibrahim Baydoun1 1ESPCI ParisTech, PSL Research University, CNRS, Univ Paris Diderot, Sorbonne Paris Cit´e, Institut Langevin, 1 rue Jussieu, F-75005, Paris, France January 17, 2018 Abstract We present a new method to calculate analytically the roots of the general complex polynomial of degree three. This method is based on the approach of appropriated changes of variable involving an arbitrary parameter. The advantage of this method is to calculate the roots of the cubic polynomial as uniform formula using the standard convention of the square and cubic roots. In contrast, the reference methods for this problem, as Cardan-Tartaglia and Lagrange, give the roots of the cubic polynomial as expressions with case distinctions which are incorrect using the standar convention. -
The Evolution of Equation-Solving: Linear, Quadratic, and Cubic
California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2006 The evolution of equation-solving: Linear, quadratic, and cubic Annabelle Louise Porter Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Mathematics Commons Recommended Citation Porter, Annabelle Louise, "The evolution of equation-solving: Linear, quadratic, and cubic" (2006). Theses Digitization Project. 3069. https://scholarworks.lib.csusb.edu/etd-project/3069 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. THE EVOLUTION OF EQUATION-SOLVING LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degre Master of Arts in Teaching: Mathematics by Annabelle Louise Porter June 2006 THE EVOLUTION OF EQUATION-SOLVING: LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino by Annabelle Louise Porter June 2006 Approved by: Shawnee McMurran, Committee Chair Date Laura Wallace, Committee Member , (Committee Member Peter Williams, Chair Davida Fischman Department of Mathematics MAT Coordinator Department of Mathematics ABSTRACT Algebra and algebraic thinking have been cornerstones of problem solving in many different cultures over time. Since ancient times, algebra has been used and developed in cultures around the world, and has undergone quite a bit of transformation. This paper is intended as a professional developmental tool to help secondary algebra teachers understand the concepts underlying the algorithms we use, how these algorithms developed, and why they work. -
Theory of Equations
CHAPTER I - THEORY OF EQUATIONS DEFINITION 1 A function defined by n n-1 f ( x ) = p 0 x + p 1 x + ... + p n-1 x + p n where p 0 ≠ 0, n is a positive integer or zero and p i ( i = 0,1,2...,n) are fixed complex numbers, is called a polynomial function of degree n in the indeterminate x . A complex number a is called a zero of a polynomial f ( x ) if f (a ) = 0. Theorem 1 – Fundamental Theorem of Algebra Every polynomial function of degree greater than or equal to 1 has atleast one zero. Definition 2 n n-1 Let f ( x ) = p 0 x + p 1 x + ... + p n-1 x + p n where p 0 ≠ 0, n is a positive integer . Then f ( x ) = 0 is a polynomial equation of nth degree. Definition 3 A complex number a is called a root ( solution) of a polynomial equation f (x) = 0 if f ( a) = 0. THEOREM 2 – DIVISION ALGORITHM FOR POLYNOMIAL FUNCTIONS If f(x) and g( x ) are two polynomial functions with degree of g(x) is greater than or equal to 1 , then there are unique polynomials q ( x) and r (x ) , called respectively quotient and reminder , such that f (x ) = g (x) q(x) + r (x) with the degree of r (x) less than the degree of g (x). Theorem 3- Reminder Theorem If f (x) is a polynomial , then f(a ) is the remainder when f(x) is divided by x – a . Theorem 4 Every polynomial equation of degree n ≥ 1 has exactly n roots. -
How Cubic Equations (And Not Quadratic) Led to Complex Numbers
How Cubic Equations (and Not Quadratic) Led to Complex Numbers J. B. Thoo Yuba College 2013 AMATYC Conference, Anaheim, California This presentation was produced usingLATEX with C. Campani’s BeamerLATEX class and saved as a PDF file: <http://bitbucket.org/rivanvx/beamer>. See Norm Matloff’s web page <http://heather.cs.ucdavis.edu/~matloff/beamer.html> for a quick tutorial. Disclaimer: Our slides here won’t show off what Beamer can do. Sorry. :-) Are you sitting in the right room? This talk is about the solution of the general cubic equation, and how the solution of the general cubic equation led mathematicians to study complex numbers seriously. This, in turn, encouraged mathematicians to forge ahead in algebra to arrive at the modern theory of groups and rings. Hence, the solution of the general cubic equation marks a watershed in the history of algebra. We shall spend a fair amount of time on Cardano’s solution of the general cubic equation as it appears in his seminal work, Ars magna (The Great Art). Outline of the talk Motivation: square roots of negative numbers The quadratic formula: long known The cubic formula: what is it? Brief history of the solution of the cubic equation Examples from Cardano’s Ars magna Bombelli’s famous example Brief history of complex numbers The fundamental theorem of algebra References Girolamo Cardano, The Rules of Algebra (Ars Magna), translated by T. Richard Witmer, Dover Publications, Inc. (1968). Roger Cooke, The History of Mathematics: A Brief Course, second edition, Wiley Interscience (2005). Victor J. Katz, A History of Mathematics: An Introduction, third edition, Addison-Wesley (2009). -
History of Mathematics Mathematical Topic Ix Cubic Equations and Quartic Equations
HISTORY OF MATHEMATICS MATHEMATICAL TOPIC IX CUBIC EQUATIONS AND QUARTIC EQUATIONS PAUL L. BAILEY 1. The Story Various solutions for solving quadratic equations ax2 + bx + c = 0 have been around since the time of the Babylonians. A few methods for attacking special forms of the cubic equation ax3 +bx2 +cx+d = 0 had been investigated prior to the discovery and development of a general solution to such equations, beginning in the fifteenth century and continuing into the sixteenth century, A.D. This story is filled with bizarre characters and plots twists, which is now outlined before describing the method of solution. The biographical material here was lifted wholesale from the MacTutor History of Mathematics website, and then editted. Other material has been derived from Dunham’s Journey through Genius. 1.1. Types of Cubics. Zero and negative numbers were not used in fifteenth cen- tury Europe. Thus, cubic equations were viewed to be in different types, depending on the degrees of the terms and there placement with respect to the equal sign. • x3 + mx = n “cube plus cosa equals number” • x3 + mx2 = n “cube plus squares equals number” • x3 = mx + n “cube equals cosa plus number” • x3 = mx2 + n “cube equals squares plus number” Mathematical discoveries at this time were kept secret, to be used in public “de- bates” and “contests”. For example, the method of depressing a cubic (eliminating the square term by a linear change of variable) was discovered independently by several people. The more difficult problem of solving the depressed cubic remained elusive. 1.2. Luca Pacioli. -
Solving the Cubic Equation
Solving the Cubic Equation Problems from the History of Mathematics Lecture 9 | February 21, 2018 Brown University A Brief History of Cubic Equations The study of cubic equations can be traced back to the ancient Egyptians and Greeks. This is easily seen in Greece and appears in their attempts to duplicate the cube and trisect arbitrary angles. As reported by Eratosthenes, Hippocrates of Chios (c. 430 BC) reduced duplicating the cube to a problem in mean proportionals: a : b = b : c = c : 2a: This idea would later inspire Menaechmus' work (c. 350 BC) in conics. The Greeks were aware of methods to solve certain cubic equations using intersecting conics,1 but did not consider general cubic equations because their framework was too influenced by geometry. 1One solution is due to Apollonius (c. 250-175 BC) and uses a hypoerbola. 1 A Brief History of Cubic Equations The Greek method of intersecting conics was completed by Islamic mathematicians. For example, Omar Khayyam (1048-1131) gave 1. examples of cubic equations with more than one solution 2. a conjecture that cubics could not be solved with ruler and compass 3. geometric solutions to all forms of cubics using intersecting conics While the techniques were geometric, the motivation was numeric: the goal in mind was to numerically approximate the roots of cubics.2 These ideas were taken back to Europe by Fibonacci. In one example, Fibonacci computed the positive solution to x3 + 2x2 + 10x = 20 to 8 decimal places (although he gives his solution in sexagesimal). 2Work done resembles Ruffini’s method. 2 Cubics in Renaissance Italy Cubics in Renaissance Italy The first real step towards the full algebraic solution of the cubic came from an Italian mathematician named Scipione del Ferro (1465-1526), who solved the depressed cubic3 x3 + px = q: Scipione del Ferro kept his method secret until right before his death. -
Unit 3 Cubic and Biquadratic Equations
UNIT 3 CUBIC AND BIQUADRATIC EQUATIONS Structure 3.1 Introduction Objectives 3.2 Let Us Recall Linear Equations Quadratic Equations 3.3 Cubic Equations Cardano's Solution Roots And Their Relation With Coefficients 3.4 Biquadratic Equations Ferrari's Solution Descartes' Solution Roots And Their Relation With Coeftlcients 3.5 Summary - 3.1 INTRODUCTION In this unit we will look at an aspect of algebra that has exercised the minds of several ~nntllematicians through the ages. We are talking about the solution of polynomial equations over It. The ancient Hindu, Arabic and Babylonian mathematicians had discovered methods ni solving linear and quadratic equations. The ancient Babylonians and Greeks had also discovered methods of solviilg some cubic equations. But, as we have said in Unit 2, they had llol thought of complex iiumbers. So, for them, a lot of quadratic and cubic equations had no solutioit$. hl the 16th century various Italian mathematicians were looking into the geometrical prob- lein of trisectiiig an angle by straight edge and compass. In the process they discovered a inethod for solviilg the general cubic equation. This method was divulged by Girolanlo Cardano, and hence, is named after him. This is the same Cardano who was the first to iiilroduce coinplex numbers into algebra. Cardano also publicised a method developed by his contemporary, Ferrari, for solving quartic equations. Lake, in the 17th century, the French mathematician Descartes developed another method for solving 4th degree equrtioiis. In this unit we will acquaint you with the solutions due to Cardano, Ferrari and Descartes. Rut first we will quickly cover methods for solving linear and quadratic equations. -
Reflection Theorems for Number Rings
Reflection theorems for number rings Evan O’Dorney July 13, 2021 Contents I Introduction 4 1 Introduction 4 1.1 Historicalbackground . ............ 4 1.2 Methods......................................... ....... 5 1.3 Results......................................... ........ 6 1.4 Outlineofthepaper ............................... .......... 7 1.5 Acknowledgements ................................ .......... 7 2 Examples for the lay reader 7 2.1 Reflectionforquadraticequations. ................ 7 2.2 Reflectionforcubicequations . .............. 9 2.3 Reflection for 2 n n boxes.................................... 11 2.4 Reflectionforquarticequations× × . ............... 12 3 Notation 15 II Galois cohomology 15 4 Étale algebras and their Galois groups 16 4.1 Étalealgebras................................... .......... 16 4.2 TheGaloisgroupofanétalealgebra . .............. 17 4.3 Resolvents...................................... ......... 17 4.4 Subextensionsandautomorphisms . .............. 18 4.5 Torsors ......................................... ....... 18 arXiv:2107.04727v1 [math.NT] 10 Jul 2021 4.6 AfreshlookatGaloiscohomology . ............. 20 5 Extensions of Kummer theory to explicitize Galois cohomology 23 5.1 TheTatepairingandtheHilbertsymbol. ............... 27 6 Rings over a Dedekind domain 29 6.1 Indicesoflattices............................... ............ 29 6.2 Discriminants................................... .......... 30 6.3 Quadraticrings.................................. .......... 32 6.4 Cubicrings ..................................... -
View This Volume's Front and Back Matter
i i “IrvingBook” — 2013/5/22 — 15:39 — page i — #1 i i 10.1090/clrm/043 Beyond the Quadratic Formula i i i i i i “IrvingBook” — 2013/5/22 — 15:39 — page ii — #2 i i c 2013 by the Mathematical Association of America, Inc. Library of Congress Catalog Card Number 2013940989 Print edition ISBN 978-0-88385-783-0 Electronic edition ISBN 978-1-61444-112-0 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “IrvingBook” — 2013/5/22 — 15:39 — page iii — #3 i i Beyond the Quadratic Formula Ron Irving University of Washington Published and Distributed by The Mathematical Association of America i i i i i i “IrvingBook” — 2013/5/22 — 15:39 — page iv — #4 i i Council on Publications and Communications Frank Farris, Chair Committee on Books Gerald M. Bryce, Chair Classroom Resource Materials Editorial Board Gerald M. Bryce, Editor Michael Bardzell Jennifer Bergner Diane L. Herrmann Paul R. Klingsberg Mary Morley Philip P. Mummert Mark Parker Barbara E. Reynolds Susan G. Staples Philip D. Straffin Cynthia J Woodburn i i i i i i “IrvingBook” — 2013/5/22 — 15:39 — page v — #5 i i CLASSROOM RESOURCE MATERIALS Classroom Resource Materials is intended to provide supplementary class- room material for students—laboratory exercises, projects, historical in- formation, textbooks with unusual approaches for presenting mathematical ideas, career information, etc. 101 Careers in Mathematics, 2nd edition edited by Andrew Sterrett Archimedes: What Did He Do Besides Cry Eureka?, Sherman Stein Beyond the Quadratic Formula, Ronald S.