Factoring and Greatest Common Factor (GCF)

Introduction to Factoring

• Factoring involves breaking down expressions into simpler parts.
• Focus of lecture: Factoring the Greatest Common Factor (GCF).

Example 1: Factoring GCF

• Expression: 3x + 15
  • Both terms are divisible by 3, hence GCF is 3.
  • Factored form: 3(x + 5)

Example 2: Factoring Binomials

• Expression: 7x - 28
  • Both terms are divisible by 7, hence GCF is 7.
  • Factored form: 7(x - 4)

More Examples

• Expression: 4x² + 8x

  • Divisible by 4x, hence GCF is 4x.
  • Factored form: 4x(x + 2)

• Expression: 5x² - 15x³

  • Divisible by 5x².
  • Factored form: 5x²(1 - 3x)

Checking Your Work

• Multiply back to ensure you get the original expression.

Factoring by Grouping

Explanation and Example

• Used for polynomials with four terms.
• Expression: 1x³ - 4x² + 3x - 12
  • Group terms: (x³ - 4x²) + (3x - 12)
  • GCF for each part: x²(x - 4) + 3(x - 4)
  • Factored form: (x - 4)(x² + 3)

Additional Example

• Expression: 2r³ - 6r² + 5r - 15
  • Group terms: (2r³ - 6r²) + (5r - 15)
  • GCF for each part: 2r²(r - 3) + 5(r - 3)
  • Factored form: (r - 3)(2r² + 5)

Factoring Trinomials

When Leading Coefficient is 1

• Expression: x² + 7x + 12
  • Find two numbers multiplying to 12 and adding to 7: 3, 4
  • Factored form: (x + 3)(x + 4)

Additional Examples

• Expression: x² + 3x - 28

  • Pair: 7, -4
  • Factored form: (x + 7)(x - 4)

• Expression: x² - 3x - 10

  • Pair: -5, 2
  • Factored form: (x - 5)(x + 2)

Factoring with Leading Coefficient Not Equal to 1

Explanation

• Multiply leading coefficient by constant: 2(-3) = -6
• Find numbers multiplying to -6 and adding to middle term.

Example

• Expression: 15x² + x - 6
  • GCF pairs: 9, -10
  • Factored form: (5x - 3)(3x + 2)

Factoring Perfect Square Trinomials

Explanation

• Form: a² + 2ab + b² = (a + b)²

Example

• Expression: x² + 8x + 16
  • Factored form: (x + 4)²

Difference of Squares

Formula

• a² - b² = (a + b)(a - b)

Examples

• Expression: x² - 25
  • Factored form: (x + 5)(x - 5)

Sums and Differences of Cubes

Formulas

• Sum of Cubes: a³ + b³ = (a + b)(a² - ab + b²)
• Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²)

Example

• Expression: x³ + 8
  • Factored form: (x + 2)(x² - 2x + 4)

Solving Equations by Factoring

Explanation

• Use zero product property after factoring.

Example

• Expression: 6x² - 30x = 0
  • GCF: 6x
  • Solved for x: x = 0 or x = 5

Additional Example

• Expression: x² - 5x - 36 = 0
  • Factor and solve.
  • Solution: x = -4 or x = 9

These notes cover the essential techniques for factoring different types of algebraic expressions, including finding the greatest common factor, factoring by grouping, factoring trinomials, perfect square trinomials, difference of squares, and solving equations by factoring.