Factoring Polynomials Using the Greatest Common Factor (GCF)

Introduction

• Goal: Learn to factor a polynomial by factoring out the Greatest Common Factor (GCF).
• Purpose of Factoring: Useful in solving polynomial equations algebraically.

Understanding the GCF

• Definition: The GCF is the largest monomial that is a factor of each term in the polynomial.
• Factoring Process: Writing an expression as an equivalent product.

Steps in Factoring with the GCF

1. Write each term in Prime Factored Form: Break down each term into its prime factors.
2. Identify Common Factors: Find the factors common to each term.
3. Factor Out the GCF: Use the distributive property to factor out the GCF, rewriting the polynomial as a product.

Example 1: Factoring 6x + 15

• Prime Factorization:
  • 6x = 2 x 3 x x
  • 15 = 3 x 5
• Identify Common Factors: Both terms have a factor of 3.
• GCF: 3
• Factored Form: 3(2x + 5)

Example 2: Factoring 2x^4 - 16x^3

• Prime Factorization:
  • 2x^4 = 2 x x x x
  • 16x^3 = 2 x 8 x x x
• Identify Common Factors: 2 and three factors of x (x^3).
• GCF: 2x^3
• Factored Form: 2x^3(x - 8)

Process for Multiple Terms

• Example: Factoring terms of 4x^2, 20x, and 12xy
  • Write each term in prime factored form.
  • Identify common factors among all terms.
  • GCF: 4xy
  • Remaining Expression: x + 5y + 3

Handling Negative Leading Coefficients

• Example: Negative leading coefficients require factoring out a negative GCF.
  • Factor out the negative GCF to change the sign of each term.
  • Check the signs to ensure accuracy.

Example 3: Factoring with Negative GCF

• Expression: Terms with no common factor other than 1 or -1.
• Action: Factor out a negative one if the leading coefficient is negative.
• Result:
  • Factored Form: Negative of the original terms with sign changes.

Conclusion

• Summary: The method provides a systematic approach to identify and factor out the GCF of polynomials.
• Outcome: Understanding of how to rewrite polynomials in factored form using the GCF.
• Gratitude: Thank you for watching and learning about factoring GCFs in polynomials.