Algebra II/Factoring Rules

1. GCF

Always take out the greatest common factor first!

• 6x²y³ - 10xy⁵ = 2xy³(3x - 5y²)
• (x+y)a⁴b⁹c + (x+y)a²bc⁵ = (x+y)a²bc(a²b⁸ + c⁴)

2. Difference of two squares: a² - b² = (a+b)(a-b)

• 9x² - 49y² = (3x+7y)(3x-7y)
• 121x² - 64 = (11x - 8)(11x + 8)
• 3x² - 12 = 3(x² - 4) → 3(x+2)(x-2)
• 3h² - 48 = 3(h² - 16) → 3(h-4)(h+4)

3. Trinomial whose leading coefficient is one

• 3x² - 18x + 24 = 3(x² - 6x + 8) → 3(x-4)(x-2)
• a²b² + 4ab - 96 = (ab-8)(ab+12)

4. Sum of two cubes: a³ + b³ = (a+b)(a² - ab + b²)

• x³ + 125 = (x+5)(x² - 5x + 25)
• 27x³ + 8 = (3x+2)(9x² - 6x + 4)
• 1000c³ - 343b³ = (10c - 7b)(100c² + 70bc + 49b²)
• 8x³ - 729 = (2x - 9)(4x² + 10x + 81)

5. Perfect Square Trinomials: a² + 2ab + b² = (a+b)²

• 9x² + 30x + 25 = (3x + 5)(3x + 5)
• 36c² + 132cd + 121d² = (6c+11d)(6c+11d)

6. Factor by Grouping

• b³ - 3b² + 4b - 12 = (b-3)(b² + 4)
• cm³ - dm³ - 8c + 8d = m³(c-d) - 8(c-d) → (m-2)(m² + 2m + 4)(c–d)

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