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Comprehensive Algebra Lecture Notes

May 7, 2025

Algebra Lecture Notes

Overview

  • Lecture notes for Algebra course at Lamar University.
  • Notes are self-contained and do not reference any book.
  • Intended for anyone needing a refresher or new learners.
  • Not a substitute for attending class; insights and additional material are often covered in class.

Preliminaries

  • Exponents: Review of integer and rational exponents, common mistakes.
  • Radicals: Definition, properties, simplified radical form.
  • Polynomials: Basics, degree, operations (add, subtract, multiply).
  • Factoring Polynomials: Greatest common factor, grouping, quadratics.
  • Rational Expressions: Reduction, operations (add, subtract, multiply, divide).
  • Complex Numbers: Standard form, operations (add, subtract, multiply, divide).

Solving Equations and Inequalities

  • Linear and Quadratic Equations: Processes for solving, applications.
  • Polynomial and Rational Inequalities: Solving strategies.
  • Absolute Value: Equations and inequalities, geometric and mathematical definitions.

Graphing and Functions

  • Graphing: Cartesian coordinate system, lines, circles.
  • Functions: Definition, notation, domain, range, graphing.
  • Combining and Inverse Functions: Operations and finding inverses.

Common Graphs

  • Parabolas, Ellipses, Hyperbolas: Graphing techniques, symmetry.
  • Transformations: Shifts, reflections.
  • Rational Functions: Graphing, asymptotes.

Polynomial Functions

  • Dividing Polynomials: Synthetic division, Division Algorithm.
  • Zeroes/Roots: Fundamental Theorem of Algebra, Factor Theorem.
  • Graphing Polynomials: Behavior at x-intercepts, leading coefficient test.
  • Partial Fractions: Decomposition of rational expressions.

Exponential and Logarithm Functions

  • Exponential Functions: Properties, applications.
  • Logarithm Functions: Properties, evaluation, change of base formula.
  • Solving Equations: Exponential and logarithmic.

Systems of Equations

  • Linear Systems: Solving with two and three variables.
  • Augmented Matrices: Gauss-Jordan elimination.
  • Nonlinear Systems: Solving strategies.

Important Notes

  • Notes include material beyond class coverage; verify with class notes.
  • Using notes as a substitute for class attendance is not recommended.