Factoring Techniques

Greatest Common Factor (GCF)

• Definition: The largest number that divides all terms in the expression.
• Example 1: For the expression 3x + 15, the GCF is 3.
  • Factor: 3(x + 5)
• Example 2: For 7x - 28, the GCF is 7.
  • Factor: 7(x - 4)

Factoring by Grouping

• Used for: Polynomials with four terms.
• Process:
  1. Separate into two groups.
  2. Factor out the GCF from each group.
  3. If both groups contain a common binomial factor, factor it out.
• Example: x^3 - 4x^2 + 3x - 12
  • Group: (x^3 - 4x^2) + (3x - 12)
  • Factor: x^2(x - 4) + 3(x - 4)
  • Final: (x - 4)(x^2 + 3)

Factoring Trinomials

• Leading coefficient (a) is 1:
  • Example: x^2 + 7x + 12
    • Numbers multiplying to 12 and adding to 7: 3, 4.
    • Factor: (x + 3)(x + 4)
• When a is not 1:
  • Example: 2x^2 + 20x + 48
    • Remove the GCF first: 2(x^2 + 10x + 24)
    • Factor: 2(x + 4)(x + 6)

Factoring Perfect Square Trinomials

• Form: a^2 + 2ab + b^2
• Example: x^2 + 8x + 16
  • Factor: (x + 4)^2

Difference of Squares

• Form: a^2 - b^2 = (a + b)(a - b)
• Example: x^2 - 25
  • Factor: (x + 5)(x - 5)

Sums and Differences of Cubes

• Formulas:
  • Sum: a^3 + b^3 = (a + b)(a^2 - ab + b^2)
  • Difference: a^3 - b^3 = (a - b)(a^2 + ab + b^2)
• Examples
  • x^3 + 8: Factor to (x + 2)(x^2 - 2x + 4)
  • 8x^3 - 27: Factor to (2x - 3)(4x^2 + 6x + 9)

Solving Equations by Factoring

• Process:
  1. Factor the equation.
  2. Use zero-product property to find solutions.
  • Example: 6x^2 - 30x = 0
    • Factor: 6x(x - 5) = 0
    • Solutions: x = 0, x = 5

Additional Notes

• Perfect Square Trinomials: Check by comparing middle term to 2ab.
• Difference of Squares and Cubes: Only difference can be factored; sums of squares cannot.