Factoring Polynomials Lecture Notes

Overview

• Discussed various methods for factoring polynomials:
  • Taking out the Greatest Common Factor (GCF)
  • Difference of Perfect Squares
  • Sum and Difference of Perfect Cubes
  • Factoring Trinomials using Substitution
  • Factoring by Grouping
  • Completing the Square using Synthetic Division
  • Solving difficult problems at the end

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Factoring Basics

Example: Factoring a Binomial

• Expression: 7x + 21
• GCF: 7
  • 7x ÷ 7 = x
  • 21 ÷ 7 = 3
• Factored Form: 7(x + 3)

Practice Problems:

1. Factor 8x² + 12xy²
   • GCF: 4xy
   • Result: 4xy(2x + 3y)
2. Factor 36x³y² - 60x⁴y³
   • GCF: 12x³y²
   • Result: 12x³y²(3 - 5xy)

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Difference of Perfect Squares

Example: Factoring x² - 25

• Formula: a² - b² = (a + b)(a - b)
• Factored Form: (x + 5)(x - 5)

Practice Problems:

1. Factor y² - 64
   • Result: (y + 8)(y - 8)
2. Factor 8x² - 18
   • GCF: 2
   • Result: 2(4x² - 9) = 2(2x + 3)(2x - 3)

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Sum and Difference of Perfect Cubes

Example: Factoring x³ + 8

• Formula: a³ + b³ = (a + b)(a² - ab + b²)
• a = x, b = 2
• Factored Form: (x + 2)(x² - 2x + 4)

Example: Factoring y³ - 125

• Formula for Difference: a³ - b³ = (a - b)(a² + ab + b²)
• Result: (y - 5)(y² + 5y + 25)

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Factoring Trinomials Using Substitution

Example: Factor x⁴ + 7x² + 12

• Substitute: a = x²
• Resulting trinomial: a² + 7a + 12
• Factored Form: (a + 3)(a + 4) → (x² + 3)(x² + 4)

Practice Problems:

1. Factor -2x⁶ + 6x³ + 56
   • Substitute: a = x³
   • Result: -2(a² - 3a - 28)

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Factoring by Grouping

Example: x³ - 2x² - 5x + 6

• Possible factors: ±1, ±2, ±3, ±6
• Using synthetic division with 1, get x² - x - 6 = (x + 2)(x - 3)
• Final Result: (x - 1)(x + 2)(x - 3)

Another Example: 4x³ - 8x² + 3x - 6

• Grouping shows: (x - 2)(4x² + 3)

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Completing the Square

Example: x² + 6x + 7

• Half of 6 is 3. Square it: 9.
• Adjust the equation: (x + 3)² - 2

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Advanced Factoring Techniques

Example: x² - 2xy + y² - 9

• Factor as (x - y)² - 9 → (x - y + 3)(x - y - 3)

Example: x²y² - y² - z² + x²z²

• Rearranging helps factor as (x² - 1)(y² + z²)
• Final Form: (x + 1)(x - 1)(y² + z²)

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Summary

• Covered a wide range of techniques for factoring polynomials.
• Practice problems provided for each technique to reinforce learning.