Polynomials Of One Variable: The Theory Of Equations
- Publish Date: 2016-03-17
- Binding: Paperback
- Author: Chris K Caldwell
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This text is designed with secondary school teachers in mind. It builds on the algebra that they regularly teach and then expand their knowledge of algebra, complex numbers and geometry. The text focuses on the interplay between these subjects--it is in this interaction between areas that mathematics truly comes alive! Chapter 2 introduces the complex numbers and their usual representations. This is done quickly in the hope that it is a review. Sections are devoted to Euler's Formula, DeMoivre's Theorem and the nth roots of complex numbers. Chapter 3 introduces polynomials, polynomial division and factorization. It next discusses the Remainder and Factor Theorems along with their usual application using synthetic division and conclude with a discussion of greatest common divisors and the Euclidean algorithm for polynomials. In chapter 4 studies the zeros of polynomials, beginning with the Fundamental Theorem of Algebra and a glance at the complex zeros of real polynomials. It concludes with Lagrange's Interpolation Formula. Chapter 5 introduces the Rational Zero Theorem as a method to find rational and integral zeros of polynomials, then presents ways to improve on this theorem and present Eisenstein's Irreducibility Criterion and reciprocal polynomials. Chapter 5 concludes with a discussion of Lill's amazing geometric solution of polynomial equations--a fun fusion of geometry and algebra! Chapter 6 studies the relationships between the zeros of polynomials and their coefficients, then use these relationships to transform and solve polynomial equations. It concludes with the Fundamental Theorem on Symmetric Functions and Newton's Identities. Chapter 7 presents the algebraic solutions to polynomials of degrees one through four, then discusses why there are no general algebraic solutions for polynomials with higher degrees. Chapter 8 uses what the reader has learned to solve the classical ruler and compass construction problems: proving that we cannot trisect the general angle, duplicate the cube, or construct most regular polygons; by first defining the geometric concept of constructability, then use the relationship between the Cartesian plane and the complex numbers to define the set of constructible numbers. The constructability of a point depends on the degree of a certain polynomial--so the reader can answer the classic questions of geometry by simply viewing the appropriate polynomials. All mathematics students could benefit by seeing this triumph of modern mathematics over classical geometry problems. Chapter 8 ends by showing how origami can be used to address these same problems. Chapters 9 and 10 return to zero finding in a more general context--presenting methods to count, separate, bound and then approximate the real zeros of polynomials (including Descartes' Rule and Sturm's Theorem). Chapter 11 studies the set of numbers which are the zeros of polynomials: the algebraic numbers along with rational, irrational and transcendental numbers. This text has hundreds of exercises ranging from the most basic applications to those that will require exploration and proof. One reason for this wide variety (and the increased difficulty of chapter eleven) is so that this text can be used to teach at a variety of different levels. An appendix contains answers to selected problems and the main text provides over 100 completely worked out examples. More could have been done with computers in this text. However the main purpose is to build the student's skill sets and confidence. One of the best ways to do this is to work things out for yourself, by hand and by mind. It is easy to delegate too much of the work to machine and calculators, leaving the user dazzled but uninformed. We must learn to master, not be mastered by, our technology.
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