Lecture Notes on Greatest Common Factor (GCF) and Polynomial Factoring

Key Topics

• Greatest Common Factor (GCF)
• Prime Factorization
• Monomials and Polynomials
• Standard Form of Polynomials
• Factoring Polynomials using GCF and Distributive Property

1. Understanding Greatest Common Factor (GCF)

• Definition: GCF is the greatest factor that is common among the terms.
• Prime Factors: Numbers whose only factors are 1 and the number itself (e.g., 2, 3, 5).
• Monomial: A polynomial with only one term (e.g., 6a).
• Polynomial: An expression made up of constants and variables, where the exponent must be a whole number.

GCF Characteristics:

• Greatest numerical factor common in terms.
• Variables should have the least degree.

Example:

• For 6x² and 15x⁴:
  • Factors of 6: 1, 2, 3, 6, x²
  • Factors of 15: 1, 3, 5, 15, x⁴
  • GCF = 3x²

2. Methods to Find GCF

Listing Method

• List out all factors and find the greatest one common to both terms.

Prime Factorization Method

• Break down numbers into their prime factors.
• Example:
  • 6x² = 2 x 3 x x²
  • 15x⁴ = 3 x 5 x x⁴
  • Common factor = 3, and the lowest degree variable is x²
  • Thus, GCF = 3x²

3. Finding GCF of Monomial Pairs

• Example 1: 6a and 18ab

  • 6a = 2 x 3 x a
  • 18ab = 2 x 3 x 3 x a x b
  • GCF = 6a

• Example 2: 10a and 12a²b

  • 10a = 2 x 5 x a
  • 12a²b = 2 x 2 x 3 x a² x b
  • GCF = 2a

• Example 3: Negative 8x²y and 16xy

  • GCF = 8xy

4. Factoring Polynomials Completely

Using GCF

• Factoring: Rewriting a polynomial as a product of polynomials with smaller degrees.
• Use GCF to rewrite and simplify.

Example:

• 4x² + 6x

  • GCF = 2x
  • Factored form: 2x(2x + 3)

• 3x² + 6x

  • GCF = 3x
  • Factored form: 3x(x + 2)

Detailed Solution Using Distributive Property

• Use GCF to simplify polynomial.
• Example:
  • 3x² + 6x = 3x(x + 2)

Binomial Factoring

• Factor by identifying common binomial factors.
• Example:
  • 7a(a + 3) - c(a + 3) = (a + 3)(7a - c)

Conclusion

• Understanding the process of finding GCF and factoring polynomials helps in simplifying complex expressions.
• Practice applying GCF and distributive property for efficient factoring.