Question

Simplify: 

i. (x + y)³ – (x – y)³

ii. (3a + 5b)³ – (3a – 5b)³

iii. (a + b)³ – a³ – b³

iv. p³ – (p + 1)³ 

v. (3xy – 2ab)³ – (3xy + 2ab)³

Answer 1

i. (x + y)³ – (x – y)³ 

Here, a = x + y and b = x – y 

(x + y)³ – (x – y)³ = [(x + y) – (x – y)] [(x + y)² + (x + y) (x – y) + (x – y)] …[a³ – b³ = (a – b)(a² + ab + b²)] 

= (x + y – x + y) [(x² + 2xy + y²) + (x² – y²) + (x² – 2xy + y²)] 

= 2y(x² + x² + x² + 2xy – 2xy + y² – y² + y²) 

= 2y(3x² + y²) = 6x²y + 2y³ 

ii. (3a + 5b)³ – (3a – 5b)³ 

Here, A = 3a + 5b and B = 3a – 5b 

= [(3a + 5b) – (3a – 5b)] [(3a + 5b)² + (3a + 5b) (3a – 5b) + (3a – 5b)²] …[∵ A³ – B³ = (A – B)(A² + AB + B²)] 

= (3a + 5b – 3a + 5b) [(9a² + 30ab + 25b²) + (9a² – 25b²) + (9a² – 30ab + 25b²)] 

= 10b(9a² + 9a² + 9a² + 30ab – 30ab + 25b² – 25b² + 25b²) 

= 10b(27a² + 25b²) 

= 270a²b + 250b³

iii. (a + b)³ – a³ – b³ 

= a³ + 3a²b + 3ab² + b³ – a³ – b³ 

= 3a²b + 3ab² 

iv. p³ – (p + 1)³ 

= p³ – (p³ + 3p² + 3p + 1) …[∵ (a + b)³ = a³ + 3a²b + 3ab² + b³] 

= p³ – p³ – 3p² – 3p – 1 

= – 3p² – 3p – 1 

v. (3xy – 2ab)³ – (3xy + 2ab)³ 

Here, A = 3xy – 2ab and B = 3xy + 2ab 

∴ (3xy – 2ab)³ – (3xy + 2ab)³ 

= [(3xy – 2ab) – (3xy + 2ab)] [(3xy – 2ab)² + (3xy – 2ab) (3xy + 2ab) + (3xy + 2ab)²] …[∵ A³ – B³ = (A – B) (A² + AB + B²)] 

= (3xy – 2ab – 3xy – 2ab) [(9x²y² – 12xyab + 4a²b²) + (9x²y² – 4a²b²) + (9x²y² + 12xyab + 4a²b²)] 

= (- 4ab) (9x²y² + 9x²y² + 9x²y² – 12xyab + 12xyab + 4a²b² – 4a²b² + 4a²b²) 

= (- 4ab) (27 xy² + 4a²b²) 

= -108x²y²ab – 16a³b³