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.