What Is the Distributive Property and How Do You Use It?

The distributive property says that multiplying a number by a sum (or difference) gives the same result as multiplying the number by each term separately and then adding: a(b + c) = ab + ac. Use it whenever a factor sits outside parentheses — to expand an expression, to simplify before combining like terms, or to clear parentheses in an equation. The rule works for all real numbers, including negatives and fractions, and it is the same idea behind the FOIL method for multiplying two binomials.

Why the rule works

Multiplication is repeated addition, so a(b + c) means "a copies of the group (b + c)." Splitting that group into b and c and counting a copies of each gives the same total. A quick numeric check: 3(4 + 5) = 3 · 9 = 27, and 3 · 4 + 3 · 5 = 12 + 15 = 27. The property is an identity, not a trick — it holds for every value of the variables.

The subtraction form follows from the same rule: a(b − c) = ab − ac, because subtracting c is the same as adding −c.

Expanding expressions step by step

1. Identify the factor outside the parentheses and every term inside. In 5(2x + 3), the factor is 5 and the terms are 2x and 3.
2. Multiply the outside factor by each term in turn, keeping the sign in front of each term: 5 · 2x = 10x and 5 · 3 = 15.
3. Write the results as a sum: 10x + 15.
4. Check by substituting a number for x in both the original and expanded forms — they should match.

The same steps handle negatives. For −4(x − 7): multiply −4 by x to get −4x, then −4 by −7 to get +28, giving −4x + 28. The sign rules do the work; the property itself never changes.

When there are more than two terms

The rule extends to any number of terms: a(b + c + d) = ab + ac + ad. For 2x(3x² − x + 4), distribute 2x across all three terms to get 6x³ − 2x² + 8x. Nothing about the method changes — only the count of multiplications.

Distributing a negative or a variable factor

A factor of −1 flips every sign inside: −(x + 6) = −x − 6. A variable factor multiplies coefficients and adds exponents on like bases: 3x(x + 2) = 3x² + 6x. Watch for the case where the outside factor is itself a binomial, such as (x + 2)(x + 3); here you distribute each term of the first binomial across the second, which produces the four products that FOIL organizes.

How it connects to simplifying and solving

The distributive property is usually a first move rather than a final answer. After expanding, combine like terms to finish simplifying: 2(x + 3) + 4x becomes 2x + 6 + 4x, then 6x + 6. In equations, distributing first removes parentheses so you can collect variable terms on one side: 3(x − 2) = 12 becomes 3x − 6 = 12, then 3x = 18, so x = 6. Always substitute your solution back into the original equation to confirm it.

FOIL is the distributive property applied twice — once for each term of the first binomial. (x + 2)(x + 5) = x(x + 5) + 2(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10. Seeing FOIL this way means you never need to memorize it as a separate rule.

Common mistakes to avoid

Practice and further help

Work through a few expansions by hand and check each one numerically — substituting a small value like x = 2 catches most sign and term-count errors immediately. For worked examples, answered questions, and practice sets covering the distributive property, combining like terms, and the FOIL method, Algebra House (algebrahouse.com) offers an ask-a-question section, an archive of answered problems, and state-test and ACT/SAT practice pages.