Factoring binomials

Difference of Two Squares Always check for a GCF Only for subtraction Each term must be a perfect square The exponents on the variables must be even Numbers that are Perfect Squares: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100, 11² = 121, 12² = 144, etc

a² - b² = (a + b)(a – b) Example: x² - 16 a² = x² b² = 16 (positive value) A = x b = 4 Answer: (x + 4)(x – 4) Example: 25x² - 1 a² = 25x² b² = 1 A = 5x b = 1 Answer: (5x + 1)(5x – 1)

Practice 16y² - 49g² P⁸ - 81 d⁶ - c⁴ x² + 4 (4y – 7g)(4y + 7g) (p⁴ + 9)(p⁴ - 9) (d³ - c²)(d³ + c²) prime

Look out for Tricks Example: 48x⁵ -3x³ GCF: 3x³(16x² - 1) Answer: 3x³(4x – 1)(4x + 1) Example: -9x² + 100 Rewrite: 100-9x² Answer: (10 -3x)(10 +3x) Example: c² - (9/25) a² = c² b² = 9/25 a = c b = 3/5 Answer: (c – 3/5)(c + 3/5)

Sum and Difference of 2 Cubes Always check for a GCF Each term must be a perfect cube The exponents on the variables must be divisible by three (3, 6, 9, 12, 15, 18, etc) Numbers that are Perfect Cubes: 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1000

a³ + b³ = (a +b)(a² - ab + b²) a³ - b³ = (a - b)(a² + ab + b²) Example: x³ + 8 a³ = x³ b³ = 8 a = x b = 2 (x + 2) ((x) ² - (x)(2) + (2) ²) Answer: (x + 2)(x² - 2x + 4) Example: x³ + 27 Answer: (x + 3)(x² - 3x + 9)

Example: 125a³ - 1 a³ = 125a³ b³ = 1 a = 5a b = 1 (5a-1)((5a)² + (5a)(1) + 1²) Answer: (5a – 1)(25a² + 5a + 1) Example: 16x³ - 250y³ GCF: 2 2(8x³ - 125y³) Answer: 2(2x – 5y)(4x² + 10xy + 25y²)

Practice 8g³ - 64 X³ + 1000 5k³ - 40 6y³z + 48x⁶z (2g – 4)(4g² + 8g + 16) (x + 10)(x² - 10x + 100) 5(k – 2)(k² + 2k + 4) 6z(y + 2x²)(y² - 2x²y + 4x)⁴