Beyond the Quadratic Formula

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Preface; 1. Polynomials; 2. Quadratic polynomials; 3. Cubic polynomials; 4. Complex numbers; 5. Cubic polynomials, II; 6. Quartic polynomials; 7. Higher-degree polynomials; Bibliography; Index.

A study guide to polynomials that goes beyond the familiar quadratic formula to cover cubic and quartic equations.

Ronald Irving has been a Professor of Mathematics at the University of Washington since 1981, prior to which he held postdoctoral positions at Brandeis University, the University of Chicago, and the University of California, San Diego. He became chair of the University of Washington's mathematics department in 2001, Divisional Dean of Natural Sciences in 2002, and Interim Dean of Arts and Sciences in 2006. In 2003, Irving joined the Board of Governors of the Astrophysical Research Consortium and continues to serve as the secretary-treasurer of the consortium. Since 2010, he has been a member of the external board of the Burke Museum of Natural History and Culture, one of Washington's state museums and a part of the University of Washington. Irving is a member of the Mathematical Association of America and the American Mathematical Society.

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*

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