Factoring Polynomials

Introduction

• Focus on how to factor polynomials
• Start with simple examples

Factoring Simple Expressions

• Example: 6x - 12

  • GCF: 6
  • Result: 6(x - 2)

• Example: 3x³ - 9x²

  • GCF: 3x²
  • Result: 3x²(x - 3)

• Example: 4x² - 12x

  • GCF: 4x
  • Result: 4x(x - 3)

Factoring Trinomials

• When the leading coefficient is 1:

  • Find two numbers that:
    • Multiply to the constant term
    • Add to the linear coefficient

• Example: x² + 7x + 12

  • Two numbers: 3 and 4
  • Result: (x + 3)(x + 4)

• Example: x² + 2x - 15

  • Two numbers: 5 and -3
  • Result: (x + 5)(x - 3)

• Example: 2x² - 6x - 56

  • Factor out GCF: 2
  • Resulting trinomial: x² - 3x - 28
  • Two numbers: -7 and 4
  • Result: 2(x - 7)(x + 4)

Factoring with GCF

• Example: 3x² - 18x + 24
  • GCF: 3
  • Resulting trinomial: x² - 6x + 8
  • Two numbers: -4 and -2
  • Result: 3(x - 4)(x - 2)

Difference of Perfect Squares

• Form: x² - a²
  • Example: x² - 16
    • Result: (x + 4)(x - 4)
  • Example: x² - 64
    • Result: (x + 8)(x - 8)

Special Cases with Leading Coefficient > 1

• Example: 2x² - 5x - 3

  • No GCF
  • Multiply leading coefficient and constant term: -6
  • Two numbers: -6 and 1
  • Replace and factor by grouping

• Example: 6x² + x - 15

  • Multiply leading coefficient and constant term: -90
  • Two numbers: 10 and -9
  • Factor by grouping: 3(2x - 3)(x + 5)

Factoring Polynomials with Four Terms

• Example: 3x³ - 2x² - 12x + 8
  • Group terms based on similar coefficients
  • Factor out GCF: x²(3x - 2) and -4(3x - 2)
  • Result: (3x - 2)(x² - 4)
  • Factor x² - 4 as (x + 2)(x - 2)

Summary

• Factoring involves finding GCF, grouping, and using methods for special cases like perfect squares.
• Practice and understanding the principles of factoring will prepare for more complex polynomials.
• For more example problems, check links in the description section.