Distributive Property — Definition, Formula & Examples
The Distributive Property is a rule that lets you multiply a number by a group of terms added (or subtracted) together by multiplying the number by each term separately, then combining the results.
For all real numbers , , and , the distributive property of multiplication over addition states that and .
Key Formula
Where:
- = The number or expression being multiplied by the group
- = The first term inside the parentheses
- = The second term inside the parentheses
How It Works
When you see a number or variable multiplied by an expression in parentheses, multiply it by every term inside the parentheses one at a time. Then add or subtract the products depending on the signs. This property works in both directions: you can expand into , or you can factor back into . Going from expanded form to factored form is sometimes called "factoring out" a common factor.
Worked Example
Problem: Simplify 5(2x + 3).
Distribute: Multiply 5 by each term inside the parentheses.
Simplify: Compute each product.
Answer:
Why It Matters
The distributive property is one of the most-used tools in algebra. You rely on it every time you simplify expressions, solve equations, or multiply polynomials like when you FOIL two binomials. It also appears constantly in mental math — for example, computing as .
Common Mistakes
Mistake: Multiplying only the first term inside the parentheses and forgetting the rest, e.g., writing 5(2x + 3) = 10x + 3.
Correction: You must multiply the outside factor by every term inside the parentheses. The correct result is 10x + 15.