Lecture on Factoring Polynomials

Introduction to Factoring

• Definition: Process of breaking down expressions into products of simpler expressions.
• Example: Factor 6x - 12
  • GCF: 6
  • Solution: 6(x - 2)

Factoring by Greatest Common Factor (GCF)

• Example 1: 3x³ - 9x²
  • GCF: 3x²
  • Solution: 3x²(x - 3)
• Example 2: 4x² - 12x
  • GCF: 4x
  • Solution: 4x(x - 3)

Factoring Trinomials

• When leading coefficient is 1:
  • Example: x² + 2x - 15
    • Find Factors: Multiply to -15, add to 2 (5 and -3)
    • Solution: (x + 5)(x - 3)
• When leading coefficient is not 1:
  • Example: 2x² - 6x - 56
    • GCF: 2
    • Factoring Trinomial: x² - 3x - 28
    • Find Factors: -7 and 4
    • Solution: 2(x - 7)(x + 4)

Special Cases: Difference of Perfect Squares

• Example 1: x² - 16
  • Solution: (x + 4)(x - 4)
• Example 2: x² - 64
  • Solution: (x + 8)(x - 8)

Factoring Trinomials with No GCF

• Example: 2x² - 5x - 3
  • Multiply First and Last Term: 2 * -3 = -6
  • Find Factors: -6 and 1
  • Grouping: Replace middle term and factor by grouping
  • Solution: (2x + 1)(x - 3)*

Factoring by Grouping

• Example: 3x³ - 2x² - 12x + 8
  • Ratios: 3/-2 and -12/8 are equal
  • Factor by Grouping: x²(3x - 2) - 4(3x - 2)
  • Final Solution: (3x - 2)(x² - 4)
  • Further Factor x² - 4: (x + 2)(x - 2)

Conclusion

• Summary: Reviewed methods of factoring polynomials including finding GCF, factoring trinomials, difference of perfect squares, and grouping.
• Additional Resources: Check links for more practice problems.