Factor a sum or difference of cubes

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

Sum of cubes

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

Difference of cubes

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

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 = xand let b = 2x5

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

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

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

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

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

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