Question

Factorise the following:

8x³ – (x – у)³

Answer 1

Given expression: 8x³ – (x – у)³

Step-wise calculation:

1. Recognize the expression as a difference of cubes, since 8x³ = (2x)³. 

2. Use the identity for the difference of cubes: 

a³ – b³ = (a – b)(a² + ab + b²)

3. Here, take a = 2x and b = x – y.

4. Substitute into the formula:

8x³ – (x – y)³ = 2x – (x – y)[(2x)² + (2x)(x – y) + (x – y)²]

5. Simplify each part:

2x – (x – y) = 2x – x + y,

2x – (x – y) = x + y, 

(2x)² = 4x², 

(2x)(x – y) = 2x² – 2xy, 

(x – y)² = x² – 2xy + y²

6. Add the terms inside the bracket:

4x² + (2x² – 2xy) + (x² – 2xy + y²) 

= 4x² + 2x² – 2xy + x² – 2xy + y²

= 7x² – 4xy + y²