# THEORY OF EQUATIONS

## NEW! Polynomial GCDs by Linear Algebra

Appendix to Chapter 1, (PDF 57KB)

## New in 2002

Portions of these notes are now available in PDF format. The files are located at http://servercc.oakton.edu/~barryd/theq/.

Chapter 1 College Algebra

Lesson 1

1.1 The Real Number System
1.2 A little History of Real and Complex Numbers
1.3 Complex Numbers
Lesson 2 1.4 Polynomial Arithmetic
1.5 Degree
1.6 The division Algorithm
1.7 Factors and Roots
Lesson 3 1.8 Greatest Common Divisor
1.9 Unique Factorization
1.10 Formal Differentiation of Polynomials
1.11 Test for Multiple Roots
1.12 Partial Fraction Decomposition
1.13 The Resultant
Chapter 2 Numerical Analysis

Lesson 4

2.1 Exact and Approximate Numbers
2.2 Numerical Algorithms
2.3 Maple Programming and Evaluation of Polynomials
2.4 Taylor's Series and Horner's Process
2.5 Synthetic Division

Lesson 5

2.6 Graphing
2.7 Descartes' Rule and Newton's Bound
2.8 The Bisection Method
2.9 Horner's Method
2.10 Iteration Method

Lesson 6

2.11 Newton's Method, Real Case
2.12 Newton's Method, Complex Case
2.13 The Newton-Barstow Algorithm
2.14 Other root finding algorithms
2.15 Polynomial Interpolation
Chapter 3 Topology

Lesson 7

3.1 History of the Fundamental Theorem of Algebra
3.2 Gauss' Fourth Proof
3.3 Topological Proof
3.4 Analysis Proof
Lesson 8 3.5 Another Curve Proof
3.6 Connection between the FTA and Newton's method
3.7 Where Newton's method does not converge
3.8 Real Newton's method revisited
Chapter 4: Ancient and Modern Algebra

Lesson 9

4.2 Omar Khayyam and Viete's Solutions of the cubic
4.3 History of the Cubic and Biquadratic
4.4 Algebraic solution of the cubic
4.5 Algebraic solution of the biquadratic
Lesson 10
4.6 Newton's Identities
4.7 More on Newton's Indentities
4.8 Symmetric Polynomials
4.9 Lagrange's Solution of the Biquadratic
4.10 Insolvability of the Quintic
Chapter 5 Number Theory

Lesson 11

5.1 Rational Polynomials and Algebraic Numbers
5.2 Integer Polynomials
5.3 Rational Roots and Factors
5.4 Eisenstein's Irreducibility Criterion
5.5 Hand Factoring methods
5.6 Computer Factoring
5.7 Comparison of factoring algorithms

Lesson 11A

Appendix to Chapter 5 on Polynomials over
the integers mod p.

## OLD 1999

### Organization and Instructions

The notes are contained in a series of HTML, GIF and LATEX DVI files. The title page, introduction, table of contents and references are HTML files which can be viewed with your browser or printed from your browser. The text of the individual chapters are in DVI format. You need special software to view and print DVI files, this is available on many UNIX systems. Personal TeX INC has a DVI viewer for windows called DVIscope which you can download for free. Postscript files are available upon request. Finally, some of the illustrations are GIF files which you can print from your browser and physically paste in the appropriate spots.

You may make copies of these notes as long as you include the link to this web page: www.neiu.edu/~bhdayton/theq in a prominent place, such as the title page, indicating that all copies originated at that link.

### The Files

• #### Chapter 2 (DVI 121KB) Numerical Analysis

Chapter 2 GIF file Figure 2.1

• #### Chapter 3 (DVI 69KB) Topology

Chapter 3 GIF files (see About the picture on the title page (HTML) for a summary of this chapter)

• #### Chapter 4 (DVI 93KB) Modern Algebra

Chapter 4 GIF files

• #### Chapter 6 (DVI 101KB) Analysis

Chapter 6 GIF files Chapter 6 Java Applets Addendum: Maple support for Elliptic functions in MAPLE V Release 4 & 5.