Difference of cubes: formula and examples

In this publication, we will consider one of the abbreviated multiplication formulas, namely, the decomposition of the difference of cubes into factors. We will also analyze examples of solving problems to consolidate the presented material.

Difference of cubes formula

The difference of cubes of numbers / expressions is equal to the product of their difference by the incomplete square of their sum.

a³ – b³ = (a – b)(a² +ab+b²)

The full square of the sum looks like this: (a+b)² = A² + 2ab + b². In our case, there is no multiplier in the second bracket opposite the second term 2, so the expression is incomplete.

The formula is also true in reverse:

(a – b)(a² +ab+b²) = the³ – b³

Note: a³ – b³ ≠ (a – b)³

Proof of the formula

It’s enough just to multiply the bracket (a – b) on (a² +ab+b²)to make sure the expression is true, i.e. go backwards:

(a – b)(a² +ab+b²) = the³ + a²b+ab² – and²b–ab² – b³ = A³ – b³.

Examples of tasks

Task 1

Express as a product of factors the expression: (7x)³ – 5³.

Solution

(7x)³ – 5³ = (7x – 5)((7x)² + 7x ⋅ 5 + 5²) = (7x – 5)(49x² + 35x + 25)

Task 2

Represent the expression 512x³ – 27 and³ in the form of a difference of cubes and decompose it into factors.

Solution

512x³ – 27 and³ = ((8x)³ – (3 and)³) = (8x – 3y)((8x)² + 8x ⋅ 3y + (3y)²) = (8x – 3y)(64x² + 24xy + 9y²)