Difference cube: formula and examples

In this publication, we will consider one of the abbreviated multiplication formulas for factoring the difference cube. We will also analyze in detail an example of solving the problem to consolidate the material.

Difference Cube Formula

Difference cube a и b equals a cube a minus three times the square a on b plus triple the product of the square b on a minus cube b.

(a – b)³ = A³ – 3a²b+3ab² – b³

The formula works in reverse:

a³ – 3a²b+3ab² – b³ = (a – b)³

Proof of the formula

Let’s represent the difference cube as a product:

(a – b)³ = (a – b)(a – b)(a – b).

Now we alternately perform the multiplication of brackets, taking into account the arithmetic rules:

(a – b)(a – b)(a – b) = (a – b)(a – b)² = (a – b)(a² – 2ab + b²) = the³ – 2a²b+ab² – and²b+2ab² – b³ = A³ – 3a²b+3ab² – b³.

Note: when opening the brackets, the formula of the square of the difference was used:

(a – b)² = A² – 2ab + b².

Example

Expand the expression (4x – 6y)³ of multiples.

Decision:

Let’s use the general formula, substituting our values ​​into it:

(4x – 6y)³ = (4x)³ – 3 ⋅ (4x)² ⋅ 6y + 3 ⋅ 4x ⋅ (6y)² – (6 and)³ = 64x³ – 288x²and + 432xy² + 216y³

Check:

Let’s multiply three identical brackets:

(4x – 6y)³ = (4x – 6y) (4x – 6y) (4x – 6y) = (4x – 6y) (4x – 6y)² = (4x – 6y)(16x² – 48xy + 36y²) = 64x³ – 192x²and + 144xy² – 96x²and + 288xy² + 216y³ = 64x³ – 288x²and + 432xy² + 216y³