Question

Factorise the following:

(2x + y)³ + (x + y)³

Answer 1

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

Step-wise calculation:

1. Let a = 2x + y and b = x + y.

2. The expression becomes (a³ + b³).

3. Recall the formula for sum of cubes: a³ + b³ = (a + b)(a² – ab + b²).

4. Calculate a + b = (2x + y) + (x + y) = 3x + 2y.

5. Calculate a² = (2x + y)² = 4x² + 4xy + y².

6. Calculate ab = (2x + y)(x + y)

ab = 2x² + 2xy + xy + y²

ab = 2x² + 3xy + y²

7. Calculate b² = (x + y)²

b² = x² + 2xy + y²

8. Form the second factor:

a² – ab + b² = (4x² + 4xy + y²) – (2x² + 3xy + y²) + (x² + 2xy + y²)

9. Simplify:

4x² + 4xy + y² – 2x² – 3xy – y² + x² + 2xy + y²

= (4x² – 2x² + x²) + (4xy – 3xy + 2xy) + (y² – y² + y²)

 = 3x² + 3xy + y²