Solved Examples: Difference of Cubes

Solved Example 1

Factor `x^3 - 8`

Solution

Rewrite 8 as `2^3`,

`x^3 - 2^3`

Comparing with `a^3 - b^3`, we get

`a = x` and `b = 2`

Applying the formula of difference of cubes: `a^3 - b^3 = (a - b)(a^2 + ab + b^2)`, we get

`(x - 2)(x^2 + x*2 + 2^2)`

Simplifying,

`(x - 2)(x^2 + 2x + 4)`

Solved Example 2

Factor `8a^3 - b^3`

Solution

Rewrite 8 as `2^3`,

`(2a)^3 - b^3`

Comparing with `a^3 - b^3`, we get,

`a = 2a` and `b = b`

Applying the formula `a^3 - b^3 = (a - b)(a^2 + ab + b^2)`, we get

`(2a - b)((2a)^2 + (2a)(b) + b^2)`

Simplifying,

`(2a - b)(4a^2 + 2ab + b^2)`

Solved Example 3

Factor `1 - x^3`

Solution

Rewrite 1 as `1^3` (because `1^3 = 1 * 1 * 1 = 1`).

`1^3 - x^3`

Comparing with `a^3 - b^3`, we get

`a = 1` and `b = x`

Applying the formula `a^3 - b^3 = (a - b)(a^2 + ab + b^2)`, we get

`(1 - x)(1^2 + 1*x + x^2)`

Simplifying,

`(1 - x)(1 + x + x^2)`