Multiplying the Difference of Two Expressions by Their Sum
Произведение разности и суммы двух выражений
Grade 7 Algebra — Special Products (Формулы сокращённого умножения)
1. Introduction to Special Products
In algebra, some polynomial multiplications occur so frequently that mathematicians have developed shortcut formulas for them. These are called:
(Short Multiplication Formulas)
❌ Without formula (slow):
Multiply each term by each term, combine like terms, simplify…
Takes many steps!
✅ With formula (fast):
Recognize the pattern → apply the formula → get the answer instantly!
Just one step!
Today we learn one of the most important of these formulas: the product of the difference and sum of two expressions.
2. Deriving the Formula
Let's multiply (a − b)(a + b) using the distributive property (each term by each term):
(a − b)(a + b)
= a · a + a · b − b · a − b · b
= a² + ab − ab − b²
= a² − b²
🔍 Notice something remarkable!
The middle terms +ab and −ab are opposites — they cancel each other out! This always happens when we multiply the sum and difference of the same two expressions.