Cubes: (a + b)³ and (a – b)³

Build the cube from the square. (a + b)³ = (a + b)(a + b)² = (a + b)(a² + 2ab + b²) = a³ + 3a²b + 3ab² + b³.

As a solid: a cube of edge a + b splits into

• a cube a³ and a cube b³;
• three cuboids a × a × b (3a²b);
• three cuboids a × b × b (3ab²).

Two cubes and six cuboids.

The minus version: put –b in place of b: (a – b)³ = a³ – 3a²b + 3ab² – b³. The signs alternate +, –, +, –. Unlike a square, a cube keeps the minus: (–b)³ = –b³.

Spot a cube by reading backwards. p³ + 6p²q + 12pq² + 8q³ = p³ + 3p²(2q) + 3p(2q)² + (2q)³ = (p + 2q)³. So a cube with this volume has side p + 2q.

In the same way, 8n³ – 60n²m + 150nm² – 125m³ = (2n)³ – 3(2n)²(5m) + 3(2n)(5m)² – (5m)³ = (2n – 5m)³.

With numbers: 199³ = (200 – 1)³ = 8000000 – 120000 + 600 – 1 = 7880599.