Distributive Law — Definition, Formula & Examples
The Distributive Law says that multiplying a number by a group of numbers added (or subtracted) together is the same as multiplying that number by each one separately, then adding (or subtracting) the results.
For all real numbers , , and , the distributive property of multiplication over addition states . Equivalently, . This property links multiplication and addition into a single consistent operation.
Key Formula
Where:
- = The factor being distributed (multiplied by each term inside the parentheses)
- = First term inside the parentheses
- = Second term inside the parentheses
How It Works
When you see an expression like , you distribute the to every term inside the parentheses. Multiply and separately, then combine. The result is . This works in reverse too: if you spot a common factor in terms like , you can factor it out to get . The distributive law is the reason techniques like combining like terms and factoring polynomials work.
Worked Example
Problem: Expand and simplify: 3(2x + 7)
Distribute 3 to the first term: Multiply 3 by 2x.
Distribute 3 to the second term: Multiply 3 by 7.
Combine the results: Write the two products together.
Answer:
Why It Matters
The distributive law is the foundation for solving equations, factoring expressions, and expanding polynomials throughout algebra. In everyday life, mental math shortcuts rely on it — for instance, computing as .
Common Mistakes
Mistake: Distributing to only the first term inside the parentheses, writing .
Correction: You must multiply the outside factor by every term inside the parentheses. The correct result is .