Beyond the Quadratic Formula

This book, written to enable self-study, addresses the problem of determining zeros of polynomials from their coefficients, avoiding modern abstract algebra and Galois theory. Irving (Univ. of Washington) developed this work from a course he taught to prospective and in-service secondary school teachers, and it would make welcome reading for any undergraduate interested in seeing some classical algebra that is no longer a regular part of the school curriculum. The author begins with a careful derivation of the quadratic formula to produce Cardano's formula for roots of cubic equations and Euler's formula for solving quartic equations as natural counterparts. Along the way, he constructs the various discriminants for determining the number of distinct real roots, as well as complex arithmetic from scratch. The volume culminates with a discussion of higher-order polynomial equations and a proof of the fundamental theorem of algebra. Irving weaves together the mathematics and the historical development of the methods throughout the book. Exercises form an integral part of the text and are embedded in the exposition so that the reader can be a partner in constructing the algebraic arguments."" - S.J. Colley, CHOICE Magazine""While the content does go beyond the quadratic formula, that distance is not great. The first four-fifths of the book is a historical and developmental walk through the tactics used to solve polynomials from quadratics up through degree four polynomials. The final section deals with quintic polynomials and the fundamental theorem of algebra. The level of material is generally well within the skill set of the advanced high school student, there are many formulas, although there is also real value in the historical details. For the student thinking about math as a career it is a demonstration of how mathematics has evolved over time at an uncertain pace. Equations considered impossible to solve are ""suddenly"" rendered solvable by one or more mathematicians that develop the correct approach or a dramatically different way of representing things. The best example of this is the development of the complex numbers. Originally used with reluctance, they changed an entire set of equations from those considered impossible to solve to ones that can be easily solved by modern high school students. Several exercises are embedded in the text, no solutions are included. For most people this will not be a problem as they will be able to develop the solutions on their own. This is a solid resource for high school mathematics, the material is well presented. i would be comfortable with giving it to a good student and telling them to learn it on their own and contact me if you need help."" - Charles Ashbacher, Journal of Recreational Mathematics""There is a great deal to like about this book. It is clearly written and will teach the reader a lot of mathematics that current undergraduates may rarely see. Future teachers, in particular, may find quite a lot of value here, since it clearly conveys the idea that the standard quadratic formula, which most students find boring, is really the tip of a very interesting iceberg."" - Mark Hunacek, MAA Review