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. A closing chapter offers glimpses into the theory of higher-degree polynomials, concluding with a proof of the fundamental theorem of algebra. The book also includes historical sections designed to reveal key discoveries in the study of polynomial equations as milestones in intellectual history across cultures.
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.
2. Quadratic Polynomials
3. Cubic Polynomials
4. Complex Numbers
5. Cubic Polynomials, II
6. Quartic Polynomials
7. Higher-Degree Polynomials
About the Author
Ron Irving is a mathematics professor at the University of Washington in Seattle. He was born in suburban New York City, studied mathematics and philosophy at Harvard, and received his Ph.D. in mathematics at MIT. Following a postdoctoral position at Brandeis and a National Science Foundation postdoctoral fellowship year at the University of Chicago and UC San Diego, Irving came to Seattle. He has been a visiting faculty member at UCSD and Aarhus and a member of the Institute for Advanced Study in Princeton. His research interests have ranged over several areas of algebra, including ring theory and the representation theory of Lie groups and Lie algebras.
When Irving began teaching the department’s senior algebra course for majors planning on secondary teaching careers, he developed an interest in the preparation of pre-service and in-service teachers. His work with this audience led to receipt of the university’s Distinguished Teaching Award and to his book Integers, Polynomials, and Rings.
Irving spent seven years in academic administration, serving as department chair for a year, divisional dean of natural sciences for four, and interim dean of the College of Arts and Sciences for two. During this time, he established the Summer Institute for Mathematics at UW, a six-week residential program that brings talented high school students in the Pacific Northwest to the university to share in the excitement of doing mathematics. He continues to serve as the program’s executive director. He also began the planning for a new undergraduate degree in integrated sciences designed to meet the needs of future secondary science teachers.
Since returning to the department, Irving has taught a variety of algebra courses and written this book. He has been the director of the Integrated Sciences program, completing its planning, approval, and implementation. He serves as the secretary-treasurer of the Astrophysical Research Consortium, which oversees Apache Point Observatory in the Sacramento Mountains of New Mexico, and is president of the board of the Burke Museum of Natural History and Culture.
Irving is a member of the Mathematical Association of America and the American Mathematical Society. As this book goes to press, he is beginning a second stint as department chair.