Notes on Factoring Polynomials

Overview of Factoring Techniques

• GCF (Greatest Common Factor)
• Difference of Perfect Squares
• Sum of Perfect Cubes
• Difference of Cubes
• Factoring Trinomials using Substitution
• Factoring by Grouping
• Completing the Square using Synthetic Division
• Advanced Problems

Factoring by Removing the GCF

Example 1: Factor 7x + 21

• GCF = 7
• Factor out 7:
  • 7(x + 3)

Example 2: Factor 8x² + 12xy²

• GCF = 4xy
• Result: 4xy(2x + 3y)

Example 3: Factor 36x³y² - 60x⁴y³

• GCF = 12x³y²
• Result: 12x³y²(3 - 5xy)

Difference of Perfect Squares

Definition

• a² - b² = (a + b)(a - b)

Example 4: Factor x² - 25

• Result: (x + 5)(x - 5)

Additional Examples

• Factor y² - 64: (y + 8)(y - 8)
• Factor 8x² - 18: 2(2x + 3)(2x - 3)

Factoring Trinomials

Example 5: Factor x² + 11x + 30

• Find numbers that multiply to 30 and add to 11: 5 and 6
• Result: (x + 5)(x + 6)

Example 6: Factor x² + 2x - 15

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

Leading Coefficient Not 1

• Example: 2x² - 5x - 3
  • Multiply first and last coefficients (2 * -3 = -6)
  • Find factors of -6 that add to -5: 6 and -1
  • Result: (2x + 1)(x - 3)

Factoring by Substitution

Example: Factor x⁴ + 7x² + 12

• Substitute a = x²
• Result: (x² + 3)(x² + 4)

Sum and Difference of Perfect Cubes

Formula

• a³ + b³ = (a + b)(a² - ab + b²)
• a³ - b³ = (a - b)(a² + ab + b²)

Example 7: Factor x³ + 8

• Result: (x + 2)(x² - 2x + 4)

Example 8: Factor y³ - 125

• Result: (y - 5)(y² + 5y + 25)

Factoring by Grouping

• Example: Factor 4x³ - 8x² + 3x - 6
  • Result: (x - 2)(4x² + 3)

Completing the Square

Example: Factor x² + 6x + 7

• Half of 6 is 3, square it to get 9
• Adjust: x² + 6x + 9 - 9 + 7
• Result: (x + 3)² - 2

Advanced Factoring Techniques

Factorization with Complex Numbers

• Example: x² + 4
• Result: (x + 2i)(x - 2i)

Synthetic Division

Example: Factor x³ - 2x² - 5x + 6

• Possible factors: ±1, ±2, ±3, ±6
• Result: (x - 1)(x² + 3x - 6)

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These notes summarize the key points from the lecture on factoring polynomials, covering techniques, examples, and definitions that are essential for understanding this key concept in algebra.