Question

Factorise the following:

(2x + 3y)³ – (x + y)³

Answer 1

Given expression: (2x + 3y)³ – (x + y)³

Step 1: Use the identity for difference of cubes

a³ – b³ = (a – b)(a² + ab + b²)

where a = 2x + 3y and b = x + y.

Step 2: Calculate (a – b)

(2x + 3y) – (x + y) = 2x + 3y – x – y 

(2x + 3y) – (x + y) = (2x – x) + (3y – y) 

(2x + 3y) – (x + y) = x + 2y

Step 3: Calculate (a²), (ab), and (b²)

a² = (2x + 3y)² = 4x² + 12xy + 9y²

b² = (x + y)² = x² + 2xy + y²

ab = (2x + 3y)(x + y)

ab = 2x² + 2xy + 3xy + 3y²

ab = 2x² + 5xy + 3y²

Step 4: Sum (a² + ab + b²)

4x² + 12xy + 9y² + 2x² + 5xy + 3y² + x² + 2xy + y²

= (4 + 2 + 1)x² + (12 + 5 + 2)xy + (9 + 3 + 1)y² 

= 7x² + 19xy + 13y²

Step 5: Write the factorised form

(2x + 3y)³ – (x + y)³ 

= (x + 2y)(7x² + 19xy + 13y²)