Lecture on Greatest Common Factor (GCF) and Polynomial Factorization

Introduction

• Discusses finding the Greatest Common Factor (GCF) using listing and prime factorization.
• Focus on factoring polynomials completely with common monomial factors using the distributive property.

Key Concepts

Greatest Common Factor (GCF)

• Definition: The largest factor shared by two or more terms.
• Prime Factors: Numbers whose only factors are 1 and the number itself, e.g., 2, 3, 5.

Monomial and Polynomial

• Monomial: A polynomial with only one term (prefix "mono" means one).
• Polynomial: An expression containing constants and variables combined through addition, subtraction, or multiplication.
• Standard Form: Arrangement of a polynomial in descending order of degrees.

Finding GCF by Listing and Prime Factorization

• Example 1: For 6x² and 15x⁴:

  • List factors of 6x²: 1, 2, 3, 6, x²
  • List factors of 15x⁴: 1, 3, 5, 15, x⁴
  • GCF is 3x², as it is the largest shared factor.

• Example 2: Using prime factorization for 6x² and 15x⁴:

  • Prime factorize: 2 x 3 for 6 and 3 x 5 for 15
  • GCF is 3x².

Additional Examples

• Example 1: GCF of 6a and 18ab:

  • Prime factors: 6a = 2 x 3 x a, 18ab = 2 x 3 x 3 x a x b
  • GCF is 6a (2 x 3 x a).

• Example 2: Negative 8x²y and 16xy:

  • GCF is 8xy using the shared prime factors and variables.

Factoring Polynomials

Rewriting as a Product of Polynomials

• Process: Use GCF to factor polynomials into products of smaller degree polynomials.
• Example: Factor 4x² + 6x:
  • GCF is 2x, factor to (2x)(2x + 3).

Using Distributive Property

• Example: 3x² + 6x using GCF 3x:
  • Use distributive property to factor as (3x)(x + 2).

Complex Polynomials

• Example: 7a(a + 3) - c(a + 3):
  • Factor out (a + 3) to get (a + 3)(7a - c).

Conclusion

• The lecture provided methods for finding the GCF and factoring polynomials using different strategies.
• Emphasis on both listing method and prime factorization for determining GCF.
• Encouraged practice with various examples to understand factoring.

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