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
1. 16y² - 49g² 1. (4y – 7g)(4y + 7g)
2. P⁸ - 81 2. (p´ + 9)(p´ - 9)
3. d¶ - c´ 3. (d³ - c²)(d³ + c²)
4. x² + 4 4. prime
Look out for Tricks
• Example: 48xµ -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
1. 8g³ - 64 1. (2g – 4)(4g² + 8g + 16)
2. X³ + 1000 2. (x + 10)(x² - 10x + 100)
3. 5k³ - 40 3. 5(k – 2)(k² + 2k + 4)
4. 6y³z + 48x¶z 4. 6z(y + 2x²)(y² - 2x²y + 4x)´