The book extends the high school curriculum and provides a backdrop for later study in calculus, modern algebra, numerical analysis, and complex variable theory. Exercises introduce many techniques and topics in the theory of equations, such as evolution and factorization of polynomials, solution of equations, interpolation, approximation, and congruences. The theory is not treated formally, but rather illustrated through examples. Over 300 problems drawn from journals, contests, and examinations test understanding, ingenuity, and skill. Each chapter ends with a list of hints; there are answers to many of the exercises and solutions to all of the problems. In addition, 69 "explorations" invite the reader to investigate research problems and related topics.
Reviews / Votes
From the reviews:
E.J. Barbeau
Polynomials
"This book uses the medium of problems to enable us, the readers, to educate ourselves in matters polynomial. In each section we are led, after a brief introduction, into a sequence of problems on a certain topic. If we do these successfully, we find that we have mastered the basics of the topic. If we have any difficulties, we can refer first to the hints, and, failing these, to the detailed solutions. These form an important and substantial part of the book, and often refer the reader on to the research literature. The book, like good literature, can be read successfully at different levels, and would not be out of place in any mathematician's library
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-MATHEMATICAL REVIEWS
"This is a two-faced book, and that's a good thing. One face is a set of enrichment materials for bright high school students. The other face is a fairly comprehensive textbook on algebraic properties of polynomials. . The present book is an excellent introduction to the subject for anyone, from high schooler to professional." (Allen Stenger, The Mathematical Association of America, August, 2011)
Series
Edition
1st ed. 1989. 3rd printing 2003
Language
Place of publication
Target group
Professional and scholarly
Research
Illustrations
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 27 mm
Weight
ISBN-13
978-0-387-40627-5 (9780387406275)
DOI
10.1007/978-1-4612-4524-7
Schweitzer Classification
1 Fundamentals.- 1.1 The Anatomy of a Polynomial of a Single Variable.- 1.2 Quadratic Polynomials.- 1.3 Complex Numbers.- 1.4 Equations of Low Degree.- 1.5 Polynomials of Several Variables.- 1.6 Basic Number Theory and Modular Arithmetic.- 1.7 Rings and Fields.- 1.8 Problems on Quadratics.- 1.9 Other Problems.- Hints.- 2 Evaluation, Division, and Expansion.- 2.1 Horner's Method.- 2.2 Division of Polynomials.- 2.3 The Derivative.- 2.4 Graphing Polynomials.- 2.5 Problems.- Hints.- 3 Factors and Zeros.- 3.1 Irreducible Polynomials.- 3.2 Strategies for Factoring Polynomials over Z.- 3.3 Finding Integer and Rational Roots: Newton's Method of Divisors.- 3.4 Locating Integer Roots: Modular Arithmetic.- 3.5 Roots of Unity.- 3.6 Rational Functions.- 3.7 Problems on Factorization.- 3.8 Other Problems.- Hints.- 4 Equations.- 4.1 Simultaneous Equations in Two or Three Unknowns.- 4.2 Surd Equations.- 4.3 Solving Special Polynomial Equations.- 4.4 The Fundamental Theorem of Algebra: Intersecting Curves.- 4.5 The Fundamental Theorem: Functions of a Complex Variable.- 4.6 Consequences of the Fundamental Theorem.- 4.7 Problems on Equations in One Variable.- 4.8 Problems on Systems of Equations.- 4.9 Other Problems.- Hints.- 5 Approximation and Location of Zeros.- 5.1 Approximation of Roots.- 5.2 Tests for Real Zeros.- 5.3 Location of Complex Roots.- 5.4 Problems.- Hints.- 6 Symmetric Functions of the Zeros.- 6.1 Interpreting the Coefficients of a Polynomial.- 6.2 The Discriminant.- 6.3 Sums of the Powers of the Roots.- 6.4 Problems.- Hints.- 7 Approximations and Inequalities.- 7.1 Interpolation and Extrapolation.- 7.2 Approximation on an Interval.- 7.3 Inequalities.- 7.4 Problems on Inequalities.- 7.5 Other Problems.- Hints.- 8 Miscellaneous Problems.- E.62 Zeros of z?1[(1 + z)n ? 1 ? zn].- E.63 Two trigonometric products.- E.64 Polynomials all of whose derivatives have integer zeros.- E.65 Polynomials with equally spaced zeros.- E.66 Composition of polynomials of several variables.- E.67 The Mandelbrot set.- E.68 Sums of two squares.- E.69 Quaternions.- Hint.- Answers to Exercises and Solutions to Problems.- Notes on Explorations.- Further Reading.
