Beyond the Quadratic Formula

✍ Scribed by Ronald S. Irving

• Publisher: The Mathematical Association of America
• Year: 2013
• Tongue: English
• Leaves: 245
• Series: Classroom Resource Materials
• Category: Library

✦ Synopsis

The quadratic formula for the solution of quadratic equations was discovered independently by scholars in many ancient cultures and is familiar to everyone. Less well known are formulas for solutions of cubic and quartic equations whose discovery was the high point of 16th century mathematics. Their study forms the heart of this book, as part of the broader theme that a polynomial's coefficients can be used to obtain detailed information on its roots. The book is designed for self-study, with many results presented as exercises and some supplemented by outlines for solution. The intended audience includes in-service and prospective secondary mathematics teachers, high school students eager to go beyond the standard curriculum, undergraduates who desire an in-depth look at a topic they may have unwittingly skipped over, and the mathematically curious who wish to do some work to unlock the mysteries of this beautiful subject.

✦ Table of Contents

Cover
S Title
Copyright
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
Beyond the Quadratic Formula
Council on Publications and Communications
CLASSROOM RESOURCE MATERIALS
Dedication
Preface
Contents
1 Polynomials
1.1 Definitions
1.2 Multiplication and Degree
1.3 Factorization and Roots
1.4 Bounding the Number of Roots
1.5 Real Numbers and the Intermediate Value Theorem
1.6 Graphs
2 Quadratic Polynomials
2.1 Sums and Products
2.2 Completing the Square
2.3 Changing Variables
2.4 A Discriminant
2.5 History
3 Cubic Polynomials
3.1 Reduced Cubics
3.2 Cardano’s Formula
3.3 Graphs
3.4 A Discriminant
3.5 History
4 Complex Numbers
4.1 Complex Numbers
4.2 Quadratic Polynomials and the Discriminant
4.3 Square and Cube Roots
4.4 The Complex Plane
4.5 A Geometric Interpretation of Multiplication
4.6 Euler’s and de Moivre’s Formulas
4.7 Roots of Unity
4.8 Converting Root Extraction to Division
4.9 History
5 Cubic Polynomials, II
5.1 Cardano’s formula
5.2 The Resolvent
5.3 The Discriminant
5.4 Cardano’s Formula Refined
5.5 The Irreducible Case
5.6 Viete’s Formula
5.7 The Signs of the Real Roots
5.8 History
6 Quartic Polynomials
6.1 Reduced Quartics
6.2 Ferrari’s Method
6.3 Descartes’ Method
6.4 Euler’s Formula
6.5 The Discriminant
6.6 The Nature of the Roots
6.7 Cubic and Quartic Reprise
6.8 History
7 Higher-Degree Polynomials
7.1 Quintic Polynomials
7.2 The Fundamental Theorem of Algebra
7.3 Polynomial Factorization
7.4 Symmetric Polynomials
7.5 A Proof of the Fundamental Theorem
References
Index
About the Author