**Dolciani Mathematical Expositions**

Volume: 35;
2009;
265 pp;
Hardcover

**Print ISBN: 978-0-88385-341-2
Product Code: DOL/35**

List Price: $62.00

AMS Member Price: $46.50

MAA Member Price: $46.50

**Electronic ISBN: 978-1-4704-5844-7
Product Code: DOL/35.E**

List Price: $62.00

AMS Member Price: $46.50

MAA Member Price: $46.50

# Uncommon Mathematical Excursions: Polynomia and Related Realms

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*Dan Kalman*

MAA Press: An Imprint of the American Mathematical Society

Recipient of the Mathematical Association of America's
Beckenbach Book Prize in 2012!

This text serves as a tour guide to little known
corners of the mathematical landscape, not far from the main byways of
algebra, geometry, and calculus. It is for the seasoned mathematical
traveller who has visited these subjects many times and, familiar with
the main attractions, is ready to venture abroad off the beaten
track. For the old hand and new devotee alike, this book will
surprise, intrigue, and delight readers with unexpected aspects of old
and familiar subjects. In the first part of the book all of the topics
are related to polynomials: properties and applications of Horner
form, reverse and palindromic polynomials and identities linking roots
and coefficients, among others. Topics in the second part are all
connected in some way with maxima and minima. In the final part
calculus is the focus.

#### Reviews & Endorsements

Kalman takes a deep look at several classical topics with which mathematics teachers are familiar, yet all will be surprised by the beauty and depth of those familiar topics and intrigued by the results uncovered.

-- Dan Teague, The Mathematics Teacher

# Table of Contents

## Uncommon Mathematical Excursions: Polynomia and Related Realms

- Cover Cover11
- Copyright ii3
- Title iii4
- Preface ix10
- Acknowledgments xi12
- Contents xiii14
- I: The Province of Polynomia 118
- 1 Horner's Form 724
- 2 Polynomial Potpourri 2542
- 2.1 A Perplexing Puzzle 2542
- 2.2 The Quadratic Formula, Insideout 2542
- 2.3 Solving Cubics with Curly Roots 2643
- 2.4 Chaos in Newton's Method 2845
- 2.5 Polynomial Interpolation 3350
- 2.6 Palindromials 3451
- 2.7 Marden's Theorem 4158
- 2.8 Solution to a Perplexing Puzzle 4360
- 2.9 History, References, and Additional Reading 4360

- 3 Polynomial Roots and Coefficients 4562
- 4 Solving Polynomial Equations 6582
- 4.1 Existence Questions 6683
- 4.2 Historic Solution of Cubics and Quartics 7188
- 4.3 Alternate Solutions for Cubics 7895
- 4.4 Alternate Solutions for Quartics 84101
- 4.5 Solving Quartics with Symmetry 87104
- 4.6 Quintic and Higher Degree Equations 92109
- 4.7 History, References, and Additional Reading 95112

- II: Maxministan 97114
- 5 Leveling with Lagrange: Constrained Maxima and Minima with Lagrangian Functions 101118
- 5.1 Intuitive Justifications for the Standard Approach 102119
- 5.2 The Lagrangian Function Approach 107124
- 5.3 An Example 108125
- 5.4 A Theorem 109126
- 5.5 Presentations of the Fallacy 110127
- 5.6 In Defense of the Fallacy 111128
- 5.7 Leveling the Playing Field 115132
- 5.8 History, References, and Additional Reading 117134

- 6 A Maxmini Miscellany 119136
- 6.1 Milkmaids and Elliptical Mirrors 119136
- 6.2 Getting an Angle on Rotated Conics 121138
- 6.3 Tracing the Borders of String Art Designs 126143
- 6.4 Weighing Means: Arithmetic and Geometric 129146
- 6.5 Isoperimetric Duality 131148
- 6.6 A Discontinuous Objective Function 137154
- 6.7 History, References, and Additional Reading 138155

- 7 Envelopes and the Ladder Problem 141158
- 8 Deflection on an Ellipse 159176

- Ill: The Calculusian Republic 171188
- 9 A Generalized Logarithm for Exponential-Linear Equations 175192
- 10 Envelopes and Asymptotes 193210
- 10.1 Envelope Basics 193210
- 10.2 The Traditional Definition of Envelope 194211
- 10.3 The Envelope Algorithm Rederived 197214
- 10.4 Boundary Points On Envelopes 199216
- 10.5 Intersections of Neighboring Curves 200217
- 10.6 Asymptotes 203220
- 10.7 Detecting Horizontal Asymptotes 207224
- 10.8 Asymptotic Polynomial s and Rational Functions 208225
- 10.9 History, References, and Additional Reading 210227

- 11 Derivatives Without Limits 213230
- 12 Two Calculusian Miracles 229246

- References 249266
- Index 259276
- About the Author 265282
- Back cover Back Cover1293