a(b + c) = ab + ac
Multiply a by both b and c.
Free Pre-Algebra Guide
Distributive Property
Learn how to multiply a factor across every term inside parentheses, handle positive and negative signs, and combine like terms after distributing.
The distributive property lets you remove parentheses by multiplying the factor outside the parentheses by every term inside. It is a key algebra skill because it appears in simplifying expressions, combining like terms, and solving multi-step equations.
On This Page
- What the distributive property means
- The basic rule
- Why every term must be multiplied
- Distribution with subtraction
- Negative outside factors
- Variable coefficients inside parentheses
- Combining like terms afterward
- Two distributive groups
- Worked examples
- Common mistakes
- A reliable strategy
- Interactive practice
What Is the Distributive Property?
The distributive property says that multiplication outside parentheses applies to every term inside the parentheses.
3(x + 4)
3 is the outside factor.
It multiplies the entire quantity inside the parentheses.
It multiplies the entire quantity inside the parentheses.
x and 4 are the inside terms.
Each one must be multiplied by 3.
Each one must be multiplied by 3.
The distributed result is 3x + 12.
The parentheses are no longer needed.
The parentheses are no longer needed.
Think: outside factor × first term, then outside factor × second term.
The Basic Distributive Property Rule
a(b − c) = ab − ac
The outside factor still multiplies both terms.
Example: 5(x + 2)
First product
5 · x = 5x
Second product
5 · 2 = 10
Result
5x + 10
Why You Must Multiply Every Term
One of the most common distributive-property mistakes is multiplying the outside factor by only the first term.
3(x + 4)
3x + 12
3(x + 4)
3x + 4
The 4 Is Inside the Parentheses Too
Because the 4 is part of the grouped expression, it must also be multiplied by 3. That gives 3 · 4 = 12.
Distributing Across Subtraction
When a subtraction sign appears inside the parentheses, treat the term after it as negative.
Example: 4(x − 3)
First product
4 · x = 4x
Second product
4 · (−3) = −12
Result
4x − 12
Negative Factors Outside Parentheses
A negative outside factor still multiplies every term. The sign of each product follows the usual multiplication rules for positive and negative numbers.
Example: −2(x − 5)
Multiply x
−2 · x = −2x
Multiply −5
−2 · (−5) = +10
Result
−2x + 10
Negative Times Negative Is Positive
This is why −2(x − 5) becomes −2x + 10, not −2x − 10.
Variables With Coefficients Inside Parentheses
If the term inside the parentheses already has a coefficient, multiply the outside factor by that coefficient.
Example: 3(2x + 5)
3(2x + 5) = 6x + 15
For the variable term, multiply 3 · 2x to get 6x. For the constant, multiply 3 · 5 to get 15.
The Variable Does Not Change
3 · 2x = 6x, not 6x². You are multiplying the numerical coefficients; the variable remains x.
Combining Like Terms After Distributing
Some expressions are not finished after the parentheses are removed. If the distributed expression contains like terms, combine them next.
Example: 3(x + 4) + 2x
Start
3(x + 4) + 2x
Distribute
3x + 12 + 2x
Combine x-terms
5x + 12
Do not skip directly to the final expression when practicing step by step:
distribute first, then combine like terms.
Expressions With Two Distributive Groups
If an expression contains two separate groups of parentheses, distribute through each group before combining like terms.
Example: 2(x + 3) + 3(x + 4)
First distribution
2x + 6 + 3(x + 4)
Second distribution
2x + 6 + 3x + 12
Combine variables
5x + 6 + 12
Combine constants
5x + 18
Worked Distributive Property Examples
5(x + 2)
- 5 · x = 5x.
- 5 · 2 = 10.
- Result: 5x + 10.
3(x − 4)
- 3 · x = 3x.
- 3 · (−4) = −12.
- Result: 3x − 12.
−4(x − 2)
- −4 · x = −4x.
- −4 · (−2) = +8.
- Result: −4x + 8.
2(3x + 5)
- 2 · 3x = 6x.
- 2 · 5 = 10.
- Result: 6x + 10.
4(x + 2) + 3x
- Distribute: 4x + 8 + 3x.
- Combine 4x and 3x.
- Result: 7x + 8.
−2(x + 4) + 3(x − 1)
- Expand the first group: −2x − 8 + 3(x − 1).
- Expand the second: −2x − 8 + 3x − 3.
- Combine: x − 11.
Common Distributive Property Mistakes
Multiplying Only the First Term
3(x + 4) is 3x + 12, not 3x + 4.
Losing a Negative Sign
In 4(x − 3), the second term is −3, so its product is −12.
Mishandling a Negative Outside Factor
−2(x − 5) becomes −2x + 10 because −2 · −5 is positive.
Adding Instead of Multiplying Coefficients
3(2x + 4) begins with 3 · 2x = 6x, not 5x.
Changing the Variable
3 · 2x = 6x, not 6x².
Combining Unlike Terms
After distributing, combine only terms with matching variable parts.
A Reliable Strategy for the Distributive Property
1. Find the outside factor
Identify the number or term multiplying the parentheses.
2. Multiply the first term
Multiply the outside factor by the first term inside.
3. Multiply every remaining term
Do not stop after the first product.
4. Check the signs
Use positive and negative multiplication rules carefully.
5. Remove the parentheses
Once every term has been multiplied, rewrite the expanded expression.
6. Combine like terms if possible
Simplify matching variable terms and constants only after distributing.
Quick reminder: distribute first, simplify second.
Practice the Distributive Property
Rewrite the entire expression after each algebra step, practice negative factors, and combine like terms after distributing.
Open the Distributive Property ToolRelated Math Resources
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