Algebra Lecture Notes

Introduction

• Covering common algebra concepts.
• Focus on like terms, polynomials, monomials, binomials, trinomial, and polynomial operations.

Like Terms

• Definition: Terms with the same variable part.
• Operations: Add coefficients of like terms.
  • Example: 5x + 4x = 9x
  • Example: 3x + 5x = 8x, 4y + 8y = 12y

Radical Expressions

• Like terms apply to expressions with the same root.
  • Example: 3√2 + 8√2 = 11√2
  • Example: 5√7 + 3√7 = 8√7

Polynomial Operations

• Addition/Subtraction: Combine like terms.
  • Example: 4x^2 + 9x^2 = 13x^2, 7x + 5x = 12x
• Multiplication: Distribute each term.
  • Example for monomial by trinomial: 7x * (x^2 + 2x - 3)
  • FOIL method for binomials: 3x - 4 * 2x + 7

Types of Polynomials

• Monomial: One term (e.g., 8x)
• Binomial: Two terms (e.g., 5x + 6)
• Trinomial: Three terms (e.g., x^2 + 6x + 5)
• Polynomial: Many terms

Properties of Exponents

• Multiplication: Add exponents (e.g., x^3 * x^4 = x^7)
• Division: Subtract exponents (e.g., x^9 / x^4 = x^5)
• Power to a Power: Multiply exponents (e.g., (x^7)^6 = x^42)*

Solving Equations

• Single Variable Linear Equations:
  • Example: x + 4 = 9, solve for x = 5
  • Example: 3x + 5 = 11, x = 2
  • Example: 2(x - 1) + 6 = 10, solve for x
• Quadratic Equations: Factorization technique and quadratic formula
  • Example: x^2 - 5x + 6 = 0
  • Factor using difference of squares
• Complex Fractions: Simplify using Keep Change Flip method

Graphing Linear Equations

• Slope-Intercept Form: y = mx + b
  • m is the slope, b is the y-intercept
• Standard Form: Ax + By = C
  • Find x- and y-intercepts to graph.

Writing Equations

• From Slope and Point: Use point-slope form to derive other forms.
• Parallel and Perpendicular Lines: Slopes of parallel lines are equal; perpendicular lines have negative reciprocal slopes.

Additional Concepts

• Complex Solutions: Understand real vs. imaginary solutions (e.g., using i for imaginary numbers).
• Solving Systems of Equations: Using substitution or elimination methods.

Practice Problems

• Multiple examples provided within each section to reinforce concepts.

Summary

• Understanding and mastering algebraic operations and solving equations is fundamental for higher-level math courses and applications.

Tips for Success

• Practice solving different types of algebraic equations.
• Familiarize yourself with operations on polynomials and properties of exponents.
• Utilize graphing techniques to understand equations visually.