Factor a sum or difference of cubes

Learn how to factor a sum or difference of cubes using the special factoring patterns below.

Example #1: 

Factor the sum of cubes x³ + 27

First rewrite x³ + 27 so it will have the same format as a³ + b³

x³ + 27 = x³ + 3³

Let a = x and let b = 3

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

x³ + 3³ = (x + 3) × (x² - x×3 + 3²)

x³ + 3³ = (x + 3) × (x² - 3x + 9)

Check

(x + 3) × (x² - 3x + 9) = x(x² - 3x + 9) + 3(x² - 3x + 9)

(x + 3) × (x² - 3x + 9) = x³ - 3x² + 9x + 3x² - 9x + 27

(x + 3) × (x² - 3x + 9) = x³ - 3x² + 3x² +9x - 9x + 27

Everything in red is equal to 0

(x + 3) × (x² - 3x + 9) = x³ + 27

Example #2: 

Factor the difference of cubes x³ - 64

First rewrite x³ - 64 so it will have the same format as a³ - b³

x³ - 64 = x³ - 4³

Let a = x and let b = 4

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

x³ - 4³= (x - 4) × (x² + x×4 + 4²)

x³ - 4³= (x - 4) × (x² + 4x + 16)

Check

(x - 4) × (x² + 4x + 16) = x(x² + 4x + 16) - 4(x² + 4x + 16)

(x - 4) × (x² + 4x + 16) = x³ + 4x² + 16x - 4x² - 16x - 64

(x - 4) × (x² + 4x + 16) = x³ + 4x² + - 4x² + 16x - 16x - 64

Everything in red is equal to 0

(x - 4) × (x² + 4x + 16) = x³ - 64

Tricky examples showing how to factor a sum or difference of cubes

Example #3: 

Factor the difference of cubes 125x³ - 216

First rewrite 125x³ - 64 so it will have the same format as a³ - b³

125x³ - 216 = (5x)³ - (6)³

Let a = 5x and let b = 6

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

(5x)³ - (6)³ = (5x - 6)[(5x)² + 5x × 6 + 6²]

(5x)³ - (6)³ = (5x - 6)(25x² + 30x + 36)

Example #4: 

Factor the sum of cubes x⁶ + 8x¹⁵

First rewrite x⁶ + 8x¹⁵ so it will have the same format as a³ + b³

x⁶ + 8x¹⁵ = (x²)³ + (2x⁵)³

Let a = x² and let b = 2x⁵

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

(x²)³ + (2x⁵)³ = (x² + 2x⁵) × [(x²)²- x²×2x⁵ + (2x⁵)²]

(x²)³ + (2x⁵)³ = (x² + 2x⁵) × (x⁴ - 2x⁷ + 4x¹⁰)

(x²)³ + (2x⁵)³ = x²(1 + 2x³) × x⁴(1 - 2x³ + 4x⁶)

(x²)³ + (2x⁵)³ = x⁶(1 + 2x³)×(1 - 2x³ + 4x⁶)