Basic Introduction to College Algebra

Introduction

• Overview of fundamental concepts in college algebra.

Basic Operations with Exponents

Multiplication of Like Bases

• Example: x^2 * x^5 = x^7
• Rule: Add exponents when multiplying the same bases
• Explanation: x^2 = x * x, x^5 = x * x * x * x * x, combined = x^7

Division of Like Bases

• Example: x^5 / x^2 = x^3
• Rule: Subtract exponents when dividing the same bases
• Explanation: x^5 (5 x’s) / x^2 (2 x’s), cancel 2 x’s → x^3

Special Cases

• Example: x^4 / x^7 = x^-3 = 1 / x^3
• Rule: Negative exponent means reciprocal

Raising Exponent to Another Exponent

• Example: (x^3)^4 = x^12
• Rule: Multiply the exponents
• Explanation: 3 * 4 = 12

Zero Exponent Rule

• Example: x^0 = 1
• Rule: Any number raised to the power of zero is 1

Simplifying and Combining Like Terms

Example 1: 5x + 3 + 7x - 4

• Combine like terms: 5x + 7x = 12x; 3 + (-4) = -1
• Result: 12x - 1

Example 2: 3x^2 + 6x + 8 + 9x^2 + 7x - 5

• Combine like terms: 3x^2 + 9x^2 = 12x^2; 6x + 7x = 13x; 8 - 5 = 3
• Result: 12x^2 + 13x + 3

Example 3: 5x^2 - 3x + 7 - 4x^2 - 8x - 11

• Distribute negative: -4x^2 + 8x + 11
• Combine like terms: 5x^2 - 4x^2 = x^2; -3x + 8x = 5x; 7 + 11 = 18
• Result: x^2 + 5x + 18

Multiplying Polynomials

FOIL Method

• Example: (3x - 5)(2x - 6)
• First terms: 3x * 2x = 6x^2
• Outside terms: 3x * -6 = -18x
• Inside terms: -5 * 2x = -10x
• Last terms: -5 * -6 = 30
• Combine like terms: -18x + -10x = -28x
• Result: 6x^2 - 28x + 30

Expanding Binomials

• Example: (2x - 5)^2
• Square each term and apply FOIL: (2x - 5)(2x - 5)
• Result: 4x^2 - 20x + 25

Solving Linear Equations

Isolation Method

• Example: x + 6 = 11, x = 5
• Example: 4x = 8, x = 2

Combination of Multiplication and Addition

• Example: 3x + 5 = 26, x = 7
• Example: 4(2x - 7) + 8 = 20, x = 5

Solving and Graphing Inequalities

Solving

• Example: 2x + 5 > 11
• Steps: Solve as linear equation
• Graph: Use open/closed circles to show exact boundary

Compound Inequalities

• Example: -6 ≤ 2x + 5 ≤ 9
• Steps: Solve for x in three parts
• Graph: Show shading between two points

Absolute Value Expressions

Definition

• Absolute value converts all inputs to positive values.

Solving

• Example: |2x + 3| = 11, leads to 2 equations 2x + 3 = 11 and 2x + 3 = -11

Inequalities

• Example: |3x - 1| > 5, leads to combination of two inequalities: 3x - 1 > 5 or 3x - 1 < -5

Graphing Functions

Slope-Intercept Form

• Example: y = 2x - 3
• Steps: Plot y-intercept, use slope to find next points

Standard Form

• Example: 2x + 3y = 6
• Steps: Find X and Y intercepts, plot and connect

Graph Transformations

Absolute Value Functions

• Examples: y = |x|, y = -|x|
• Shifts and reflections: y = |x + 2|, y = |x| + 3

Quadratic Functions

• Standard form: y = x^2
• Transformations: y = (x - 2)^2 , shifts and reflections

Factoring Quadratic Equations

Difference of Squares

• Example: x^2 - 25 = (x - 5)(x + 5)

General Method

• Example: x^2 + 10x + 24 = (x + 4)(x + 6)

Leading Coefficient ≠ 1

• Example: 2x^2 + 5x - 12, factor to (2x + 3)(x - 4)

Quadratic Formula

• Formula: x = (-b ± √(b^2 - 4ac)) / 2a
• Example: Used to confirm above results

Solving Systems of Equations

Substitution and Elimination Methods

• Example: 2x + y = 5, 3x - y = 0

Multiple Methods

• Solving by substitution where functions are more complex

Evaluating and Graphing Functions

Composite Functions

• Example: f(g(x)) and g(f(x)) with defined functions.

Inverse Functions

• Finding inverse, checking through composition

Closing

• Encouragement to review additional algebra topics through provided playlists and additional videos in various mathematical subjects.

Additional Topics and Resources

• Mention of playlists for physics, chemistry, and calculus
• Availability of further learning materials on the lecturer’s channel