Notes on Factoring

Introduction to Factoring

• Focus on factoring the Greatest Common Factor (GCF).

Example 1: Factoring 3x + 15

• GCF of 3x and 15 is 3.
• Factoring Result:
  • 3(x + 5)

Example 2: Factoring 7x - 28

• GCF of 7x and -28 is 7.
• Factoring Result:
  • 7(x - 4)

Practice Problems

• 4x² + 8x

  • GCF: 4x
  • Factoring Result: 4x(x + 2)

• 5x² - 15x³

  • GCF: 5x²
  • Factoring Result: 5x²(1 - 3x)

Factoring by Grouping

• Useful for polynomials with four terms.
• Example: Factor x³ - 4x² + 3x - 12
  • Group into two: (x³ - 4x²) + (3x - 12)
  • GCF of first group: x² (x - 4)
  • GCF of second group: 3 (x - 4)
  • Factoring Result: (x - 4)(x² + 3)

Example: 2r³ - 6r² + 5r - 15

• Factor the first two terms: GCF = 2r²
• Factor the last two terms: GCF = 5
• Factoring Result: (r - 3)(2r² + 5)

Factoring Trinomials (Leading Coefficient = 1)

• Find two numbers that multiply to the constant term and add to the middle coefficient.
• Example: x² + 7x + 12
  • Factors: (x + 3)(x + 4)
• Example: x² + 3x - 28
  • Factors: (x + 7)(x - 4)

Factoring Trinomials (Leading Coefficient ≠ 1)

• Multiply leading coefficient by the constant term, find two numbers that fit.
• Example: 2x² - 5x - 6
  • Rewrite as: 2x² - 6x + x - 6
  • Factor by grouping.

Perfect Square Trinomials

• Recognized in the form a² + 2ab + b².
• Example: x² + 8x + 16
  • Factors: (x + 4)²

Difference of Squares

• Form: a² - b² = (a + b)(a - b).
• Example: x² - 25
  • Factors: (x + 5)(x - 5)

Sums and Differences of Cubes

• Sum of Cubes: a³ + b³ = (a + b)(a² - ab + b²)
• Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²)
• Example: x³ + 8 = (x + 2)(x² - 2x + 4)

Solving Equations by Factoring

• Use the zero product property: if ab = 0, then a = 0 or b = 0.
• Example: 6x² - 30x = 0
  • Factored: 6x(x - 5) = 0
  • Solutions: x = 0 or x = 5.

Conclusion

• Practicing different forms and methods of factoring is essential to master solving polynomials.