Algebra Lecture Notes: Factoring the Greatest Common Factor (GCF)

Introduction to Factoring

• Factoring: Process of figuring out what things multiply together to give a certain polynomial.
• Simplified Definition: Factoring is distribution in reverse.
• Importance: Essential for solving equations, particularly quadratic equations, in algebra.

Basic Principles

• Distribution: Multiply a term outside parentheses across terms inside.
• Factoring: Take a given expression and identify what can be pulled out as a common factor.

Understanding Factoring

• Example 1:

  • Expression: X + 2
  • No common factors between X and 2, therefore it cannot be factored.

• Example 2:

  • Expression: 2X + 2
  • Common factor: 2
  • Factored form: 2(X + 1)
  • Process: Identify the common factor (2) and rewrite the expression using it.

Greatest Common Factor (GCF)

• Definition: Largest factor common to each term in a polynomial.
• Example:
  • Expression: 6X + 3
  • GCF: 3
  • Factored form: 3(2X + 1)
  • Understanding Factor Trees: Break terms down into their factors to identify the GCF.

Factoring Process

1. Identify the GCF of all terms.
2. Factor it out.
3. Write the remaining expression inside parentheses.
4. Verify by distributing to ensure the original expression is obtained.

Practice Problems

• Example 1:

  • Expression: X^2 + XY
  • GCF: X
  • Factored form: X(X + Y)

• Example 2:

  • Expression: 10X^2 + 5X^2Y
  • GCF: 5X^2
  • Factored form: 5X^2(2 + Y)

• Example 3:

  • Expression: 3P^3Q + 15P^2
  • GCF: 3P^2
  • Factored form: 3P^2(PQ + 5)

Advanced Examples

• Example 4:

  • Expression: 49X^3 + 35X^2Y
  • GCF: 7X^2
  • Factored form: 7X^2(7X + 5Y)

• Example 5:

  • Expression: 25XY^2Z + 15Y^2Z^2
  • GCF: 5Y^2Z
  • Factored form: 5Y^2Z(5X + 3Z)

• Example 6:

  • Expression: 24R^2S + 36RS^2T
  • GCF: 6RS
  • Factored form: 6RS(4R + 6ST)
  • Note: Double-check factored form for any missed common factors.

Conclusion

• Key Takeaway: Factoring is critical to solve equations in algebra. It requires practice to identify and factor out the greatest common factor.
• Next Steps: Practice more complex problems and be prepared to use factoring in solving quadratic equations.

Tips for Success

• Double-Check: Always verify by redistributing.
• Common Errors: Forgetting common factors or failing to factor completely.
• Practice: Work on various problems to become proficient in identifying and factoring GCF.