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

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
3.9 Iteration of Quadratic Polynomials
Chapter 4: Ancient and Modern Algebra

Lesson 9

4.1 Solutions of Quadratic Equations
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

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The Files


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