Distributive Property
The distributive property is the rule that multiplying a number by a group of numbers added together gives the same result as multiplying the number by each one separately and then adding. In symbols, .
The distributive property of multiplication over addition states that for any real numbers , , and , the expression is equivalent to . This property also applies to subtraction: . It serves as a foundational rule in algebra, allowing you to expand expressions and factor common terms out of sums.
Key Formula
Where:
- = the factor being distributed (multiplied by each term inside the parentheses)
- = the first term inside the parentheses
- = the second term inside the parentheses
Worked Example
Problem: Simplify 4(3x + 7) using the distributive property.
Step 1: Identify the factor outside the parentheses and the terms inside.
Step 2: Multiply the outside factor by the first term inside.
Step 3: Multiply the outside factor by the second term inside.
Step 4: Add the two products together.
Answer:
Visualization
Why It Matters
The distributive property is one of the most frequently used tools in algebra. Any time you expand an expression like or factor out a common term from , you're relying on this property. It also underpins mental math shortcuts — for instance, computing by thinking of it as .
Common Mistakes
Mistake: Distributing to only one term inside the parentheses, e.g., writing .
Correction: You must multiply the outside factor by every term inside. The correct result is .
Mistake: Forgetting to distribute the sign. For example, writing .
Correction: A negative times a negative is a positive. Since , the correct expansion is .