Beyond the Quadratic Formula

(Classroom Resource Materials)

Ronald S. Irving

The Mathematical Association of America | 2013 | 244 páginas | rar - pdf | 1,1 Mb

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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.

Contents
Preface ix
1 Polynomials 1
1.1 Definitions . . 1
1.2 Multiplication and Degree. . 4
1.3 Factorization and Roots. . 8
1.4 Bounding the Number of Roots . . 10
1.5 Real Numbers and the Intermediate Value Theorem . . 12
1.6 Graphs . . 16
2 Quadratic Polynomials 21
2.1 Sums and Products . . 22
2.2 Completing the Square . . 24
2.3 Changing Variables . . 28
2.4 A Discriminant . . 29
2.5 History  . 33
3 Cubic Polynomials 47
3.1 Reduced Cubics  . . 47
3.2 Cardano’s Formula . . 50
3.3 Graphs. . 58
3.4 A Discriminant . . 61
3.5 History . . 66
4 Complex Numbers 73
4.1 Complex Numbers .  . 73
4.2 Quadratic Polynomials and the Discriminant . . 77
4.3 Square and Cube Roots . . 81
4.4 The Complex Plane . . 84
4.5 A Geometric Interpretation of Multiplication . . 88
4.6 Euler’s and de Moivre’s Formulas . . 92
4.7 Roots of Unity . . 98
4.8 Converting Root Extraction to Division . . 101
4.9 History . . 103
5 Cubic Polynomials, II 109
5.1 Cardano’s formula  . . 109
5.2 The Resolvent . . 113
5.3 The Discriminant . . 115
5.4 Cardano’s Formula Refined. . 120
5.5 The Irreducible Case. . 124
5.6 Vi`ete’s Formula . . 125
5.7 The Signs of the Real Roots . . 130
5.8 History  . . 133
6 Quartic Polynomials 143
6.1 Reduced Quartics . . 143
6.2 Ferrari’s Method  . . 146
6.3 Descartes’ Method . . 149
6.4 Euler’s Formula . . 154
6.5 The Discriminant . . 157
6.6 The Nature of the Roots . . 162
6.7 Cubic and Quartic Reprise. . 167
6.8 History  . . 169
7 Higher-Degree Polynomials 179
7.1 Quintic Polynomials  . . 179
7.2 The Fundamental Theorem of Algebra . . 185
7.3 Polynomial Factorization . . 191
7.4 Symmetric Polynomials. . 200
7.5 A Proof of the Fundamental Theorem . . 211
References 217
Index 223A
About the Author 227