Factoring Polynomials

Simple Examples

• Example 1:

  • Expression: 6x - 12
  • Factor: Take out the greatest common factor (GCF) which is 6
  • Result: 6(x - 2)

• Example 2:

  • Expression: 3x³ - 9x²
  • GCF: 3x²
  • Result: 3x²(x - 3)

• Example 3:

  • Expression: 4x² - 12x
  • GCF: 4x
  • Result: 4x(x - 3)

Factoring Trinomials

• Trinomial with leading coefficient 1:

  • Example: x² + 2x - 15
  • Find two numbers that multiply to -15 and add to 2: 5 and -3
  • Result: (x + 5)(x - 3)

• Trinomial with non-1 leading coefficient:

  • Example: 2x² - 6x - 56
  • GCF: 2
  • New trinomial: x² - 3x - 28
  • Find two numbers that multiply to -28 and add to -3: -7 and 4
  • Result: 2(x - 7)(x + 4)

Difference of Perfect Squares

• Example 1: x² - 16

  • Factor: (x + 4)(x - 4)

• Example 2: x² - 64

  • Factor: (x + 8)(x - 8)

• Example 3: 4x² - 25

  • Factor: (2x + 5)(2x - 5)

• Example 4: 9x² - 49

  • Factor: (3x + 7)(3x - 7)

Advanced Factoring Techniques

• No GCF Available Example:

  • Expression: 2x² - 5x - 3
  • Multiply leading coefficient by constant: 2 * -3 = -6
  • Find numbers that multiply to -6 and add to -5: -6 and 1
  • Factor by grouping: 2x² + 1x - 6x - 3
  • Result: (2x + 1)(x - 3)

• Complex Example:

  • Expression: 6x² + x - 15
  • Multiply leading coefficient by constant: 6 * -15 = -90
  • Find numbers that multiply to -90 and add to 1: -9 and 10
  • Factor by grouping: 3x(2x - 3) + 5(2x - 3)
  • Result: (2x - 3)(3x + 5)

Factoring with Four Terms

• Example:
  • Expression: 3x³ - 2x² - 12x + 8
  • Ratio: First two and last two coefficients have the same ratio (3/-2 = -12/8)
  • Factor by grouping: x²(3x - 2) - 4(3x - 2)
  • Rewrite: (3x - 2)(x² - 4)
  • Factor x² - 4: (x + 2)(x - 2)
  • Final Result: (3x - 2)(x + 2)(x - 2)

Conclusion

• For more examples, check the links in the video description.