The Distributive Property
The distributive property says that multiplying a sum is the same as multiplying each piece and then adding: . In practice, it's the rule for clearing parentheses — the number outside multiplies each term inside, not just the first one.
It's one of the most-used moves in all of algebra. You need it to simplify expressions like , to solve equations that have parentheses, and later to multiply polynomials. Learn to distribute cleanly — especially with negative signs — and a whole category of mistakes disappears.
What the property says
In , the multiplies each term inside the parentheses: and , so . Every term inside gets its own multiplication — the most common error is multiplying only the first term and writing .
You can see why it works with plain numbers. is , and distributing gives — same answer. Three groups of really is three 's plus three 's.
Negatives and subtraction
The sign travels with the number. In , you distribute to each term: and , so . A negative times a negative is positive, and this is exactly where students drop that rule under pressure.
Subtraction inside the parentheses works the same way — treat as and multiply each term, sign included. When the outside number is negative, expect every sign inside to flip.
Distribute first, then combine like terms
Distributing usually isn't the last step. In , distribute to get , then combine the like terms and to finish with . The order is always the same: clear the parentheses first, then combine whatever matches.
Worked examples
Example 1: a positive number outside
Distribute .
Answer:
Example 2: subtraction inside
Distribute .
Answer:
Example 3: a negative number outside
Distribute .
Answer:
Example 4: distribute, then combine like terms
Simplify .
Answer:
Try one yourself
Common questions
Do I multiply every term inside the parentheses, or just the first one?
Every term. is , not . If there are three terms inside, the outside number multiplies all three. Stopping after the first term is the single most common distributing mistake.
What happens when the number outside is negative?
The negative distributes too, so every sign inside flips. : the times gives , and the times gives . Check each product's sign one at a time.
Does it work if the number is behind the parentheses, like ?
Yes. Multiplication works in either order, so . The multiplier distributes to each term no matter which side it sits on.
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