Course Overview

This study guide covers the main topics in a standard College Algebra course, as outlined in the provided weekly schedule. The guide is organized by major algebraic concepts, with definitions, properties, examples, and key formulas to support exam preparation.

Fractions and Rational Expressions

Fraction Review

Fractions represent parts of a whole and are fundamental in algebraic manipulation.

• Definition: A fraction is an expression of the form , where and are integers and .

• Operations: Addition, subtraction, multiplication, and division of fractions require a common denominator (for addition/subtraction) and multiplication/division of numerators and denominators.

• Example:

Rational Expressions

Rational expressions are fractions where the numerator and/or denominator are polynomials.

• Definition: A rational expression is of the form , where and are polynomials and .

• Operations: Similar to numerical fractions, but require factoring and simplifying polynomials.

• Example: (for )

Polynomials

Adding, Subtracting, and Multiplying Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients.

• Definition: A polynomial is an expression like .

• Addition/Subtraction: Combine like terms.

• Multiplication: Use distributive property or special products (e.g., ).

• Example:

Special Products

• Square of a Binomial:

• Difference of Squares:

• Example:

Factoring Polynomials

Factoring is expressing a polynomial as a product of its factors.

• Greatest Common Factor (GCF): Factor out the largest common factor from all terms.

• Factoring Trinomials: For , find two numbers that multiply to and add to .

• Special Factoring Techniques: Use patterns like difference of squares, perfect square trinomials, and sum/difference of cubes.

• Example:

Linear Equations and Inequalities

Solving Linear Equations

Linear equations are equations of the first degree (highest exponent of the variable is 1).

• Standard Form:

• Solution:

• Example:

Linear Inequalities

• Definition: An inequality like .

• Solution: Solve as an equation, then consider the direction of the inequality. If multiplying/dividing by a negative, reverse the inequality sign.

• Example:

Quadratic Equations

Solving Quadratic Equations

Quadratic equations are second-degree equations of the form .

• Factoring: Express as a product of binomials and set each factor to zero.

• Quadratic Formula:

• Completing the Square: Rewrite in the form .

• Example:

Complex Numbers

• Definition: Numbers of the form , where .

• Example:

Functions and Their Graphs

Basics of Functions

A function is a relation that assigns exactly one output to each input.

• Notation:

• Domain and Range: The set of possible inputs (domain) and outputs (range).

• Example:

Linear Functions and Slope

• Equation: , where is the slope and is the y-intercept.

• Slope Formula:

• Example: The line through and has slope

Transformations of Functions

• Vertical and Horizontal Shifts: shifts up/down; shifts right/left.

• Reflections: reflects over the x-axis; reflects over the y-axis.

• Example: ; shifts right by 2 units.

Combinations and Composition of Functions

• Sum, Difference, Product, Quotient: , etc.

• Composition:

• Example: If , , then

Inverse Functions

• Definition: is the inverse of if

• Finding the Inverse: Swap and in and solve for .

• Example:

Radical Expressions and Rational Exponents

Radical Expressions

• Definition: Expressions involving roots, such as .

• Properties:

• Example:

Rational Exponents

• Definition: Exponents that are fractions, e.g.,

• Example:

Quadratic Functions and Graphs

Quadratic Functions

• Standard Form:

• Vertex:

• Axis of Symmetry:

• Example: has vertex at

Distance and Midpoint Formulas

• Distance:

• Midpoint:

• Example: Points and : ,

Polynomial and Rational Functions

Polynomial Functions

• Zeros: Solutions to

• Remainder Theorem: The remainder of divided by is

• Factor Theorem: is a factor of if

Rational Functions

• Definition:

• Asymptotes: Vertical at zeros of ; horizontal/oblique based on degrees of and

• Example: has a vertical asymptote at

Exponential and Logarithmic Functions

Exponential Functions

• Definition: , ,

• Properties:

• Example:

Logarithmic Functions

• Definition: is the inverse of

• Properties:

• Example:

Exponential Growth and Decay

• Formula: , where for growth, for decay

• Example: Population growth, radioactive decay

Systems of Equations

Linear Systems in Two and Three Variables

• Definition: A set of equations with multiple variables solved simultaneously.

• Methods: Substitution, elimination, and matrix methods.

• Example: has solution

Summary Table: Key Algebraic Concepts

Additional info: This guide is based on a standard College Algebra syllabus and includes expanded academic context for clarity and completeness.