Куб суммы и куб разности
Cubing a Sum and Difference
Формулы сокращённого умножения — Short Multiplication Formulas
7 класс · Алгебра1. Введение — Introduction
You already know the формулы сокращённого умножения (short multiplication formulas) for squaring binomials:
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
(a − b)² = a² − 2ab + b²
Now we go one step further — we raise binomials to the third power (cube them):
(a + b)³ = ? (a − b)³ = ?
These are called the куб суммы (cube of a sum) and куб разности (cube of a difference). Each formula produces four terms (not three like the square formulas).
2. Вывод формулы куба суммы — Deriving the Cube of a Sum
We start from the definition of cubing:
(a + b)³ = (a + b)² · (a + b)
Step 1: Use the square formula to expand (a + b)²:
(a + b)² = a² + 2ab + b²
Step 2: Multiply by (a + b):
(a² + 2ab + b²) · (a + b)
= a² · a + a² · b + 2ab · a + 2ab · b + b² · a + b² · b
= a³ + a²b + 2a²b + 2ab² + ab² + b³
= a² · a + a² · b + 2ab · a + 2ab · b + b² · a + b² · b
= a³ + a²b + 2a²b + 2ab² + ab² + b³
Step 3: Combine like terms:
a³ + (1 + 2)a²b + (2 + 1)ab² + b³
(a + b)³ = a³ + 3a²b + 3ab² + b³