Difference of cubes

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Definition.

Difference of cubes of two expressions can be found using the following formula:

a³ - b³ = (a - b)·(a² + ab + b²)

Derivation of the formula of difference of cubes

The proof of the formula is very simple. To prove the formula is sufficient to multiply the expression:

(a - b)·(a² + ab + b²) =

= a³ + a²b + ab² - ba² - ab² - b³ = a³ - b³

Difference of cubes formula convenient to use:

Example 1.

Factorised x³ - 27.

Solution: Apply the difference of cubes formula.

x³ - 27 = x³ - 3³ = (x - 3)·(x² + 3x + 9)

Example 2.

Factorised 8x³ - 27y⁶.

Solution: Apply the difference of cubes formula.

8x³ - 27y⁶ = (2x)³ - (3y²)³ =

= (2x - 3y²)·(4x² + 6xy² + 9y⁴)

Example 3.

Simplify the expression 27x³ - 13x - 1.

Solution: Apply the difference of cubes formula in numerator.

27x³ - 13x - 1 = (3x - 1)·(9x² + 3x +1)3x - 1 = 9x² + 3x +1

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