Question

Simplify the following:

(a + b)³ + (a – b)³

Answer 1

Given: (a + b)³ + (a – b)³

Step-wise calculation:

1. Expand (a + b)³ using the binomial expansion:

(a + b)³ = a³ + 3a²b + 3ab² + b³

2. Expand (a – b)³ similarly:

(a – b)³ = a³ – 3a²b + 3ab² – b³

3. Add the two expanded expressions:

(a + b)³ + (a – b)³

= (a³ + 3a²b + 3ab² + b³) + (a³ – 3a²b + 3ab² – b³)

4. Combine like terms:

a³ + a³ + 3a²b – 3a²b + 3ab² + 3ab² + b³ – b³ = 2a³ + 0 + 6ab² + 0

a³ + a³ + 3a²b – 3a²b + 3ab² + 3ab² + b³ – b³ = 2a³ + 6ab²

5. Factor out common terms:

2a³ + 6ab² = 2a(a² + 3b²)