Distributive Property: Rule and Examples

The distributive property says that multiplying a number by a sum is the same as multiplying it by each part and then adding: $a(b + c) = ab + ac$ . For example, $3(x + 4) = 3x + 12$ . You use the distributive property to remove parentheses and simplify expressions.

Before this lesson: You should know how to combine like terms and how to multiply negative numbers.

Extra practice: The Algebra 1 Unit 1 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.

What Is the Distributive Property?

Picture a rectangle that is 3 units tall. Its width is made of two parts: $x$ and 4.

A rectangle 3 units tall split into two parts. The left part is x wide with area 3x. The right part is 4 wide with area 12. Below it: 3(x + 4) = 3x + 12. — The whole area, 3(x + 4), equals the two smaller areas added together: 3x + 12.

You can find the area in two ways. Multiply the height by the whole width: $3(x + 4)$ . Or find each small area and add them: $3x + 12$ . Both ways give the same area.

So the number outside the parentheses gets multiplied by every term inside. We say it is distributed, or shared out, to each term.

The Distributive Property Rule

\begin{aligned} a(b + c) &= ab + ac \\ a(b - c) &= ab - ac \end{aligned}

To use it, follow these steps:

Multiply the number outside the parentheses by the first term inside.
Multiply it by every other term inside. Keep track of each sign.
Combine like terms if there are any.

The number can also be on the right: $(b + c)a = ba + ca$ . And a minus sign in front of parentheses works like multiplying by $-1$ : $-(x + 3) = -x - 3$ .

Worked Examples

Example 1: A positive number outside

Simplify $3(x + 4)$ .

3(x + 4) = 3 \cdot x + 3 \cdot 4

Step 1. Multiply the 3 by each term inside the parentheses.

3 \cdot x + 3 \cdot 4 = 3x + 3 \cdot 4

Step 2. Multiply 3 by $x$ .

3x + 3 \cdot 4 = 3x + 12

Step 3. Multiply 3 by 4.

Answer: $3x + 12$

Example 2: A negative number outside

Simplify $-2(x - 5)$ .

-2(x - 5) = (-2)(x) + (-2)(-5)

Step 1. Multiply $-2$ by each term inside. The second term is $-5$ , so keep its sign.

(-2)(x) + (-2)(-5) = -2x + (-2)(-5)

Step 2. Multiply $-2$ by $x$ .

-2x + (-2)(-5) = -2x + 10

Step 3. Multiply $-2$ by $-5$ . A negative times a negative is positive. (See multiplying and dividing integers.)

Answer: $-2x + 10$

Example 3: Three terms inside

Simplify $4(2a - 3b + 1)$ .

4(2a - 3b + 1) = 4(2a) + 4(-3b) + 4(1)

Step 1. Multiply 4 by each of the three terms.

8a + 4(-3b) + 4(1)

Step 2. Multiply 4 by $2a$ .

8a - 12b + 4(1)

Step 3. Multiply 4 by $-3b$ .

8a - 12b + 4

Step 4. Multiply 4 by 1.

Answer: $8a - 12b + 4$

Example 4: Distribute, then combine like terms

Simplify $5(x + 2) - 3(x - 4)$ .

5(x + 2) - 3(x - 4) = 5x + 10 - 3(x - 4)

Step 1. Distribute the 5.

5x + 10 - 3(x - 4) = 5x + 10 - 3x + 12

Step 2. Distribute the $-3$ . Treat the minus sign as part of the $-3$ , so $(-3)(-4) = 12$ .

5x + 10 - 3x + 12 = 5x - 3x + 10 + 12

Step 3. Move the like terms next to each other (see combining like terms).

5x - 3x + 10 + 12 = 2x + 10 + 12

Step 4. Combine the $x$ -terms.

2x + 10 + 12 = 2x + 22

Step 5. Combine the constants.

Answer: $2x + 22$

Example 5: Word problem

A garden is 5 feet wide and $x + 8$ feet long. Write its area without parentheses. Then find the area when $x = 4$ .

A = 5(x + 8)

Step 1. Area equals width times length.

A = 5x + 40

Step 2. Distribute the 5: $5 \cdot x = 5x$ and $5 \cdot 8 = 40$ .

A = 5(4) + 40

Step 3. Replace $x$ with 4.

A = 20 + 40

Step 4. Multiply.

A = 60

Step 5. Add.

Answer: The area is $5x + 40$ square feet, which is 60 square feet when $x = 4$ .

Check: $5(4 + 8) = 5(12) = 60$ .

Common Mistakes

Mistake	What it looks like	How to fix it
Multiplying only the first term	$3(x + 4) = 3x + 4$	Multiply every term: $3x + 12$ .
Losing a sign	$-2(x - 5) = -2x - 10$	$(-2)(-5) = +10$ , so the answer is $-2x + 10$ .
Ignoring a minus in front of parentheses	$8 - (x + 2) = 8 - x + 2$	The minus goes to both terms: $8 - x - 2 = 6 - x$ .
Adding instead of multiplying	$3(x + 4) = 3 + x + 4$	The number outside multiplies each term. It is not added.
Distributing only the number, not its sign	$10 - 4(y + 2)$ becomes $10 - 4y + 8$	Distribute $-4$ : $10 - 4y - 8 = -4y + 2$ .

Quick check: Choose a value for the variable. The original expression and your answer should give the same number.

Other Ways to Use the Distributive Property

Mental math

Split a hard number into easier parts. To find 7 times 23, think of 23 as 20 + 3.

7(23) = 7(20 + 3) = 140 + 21 = 161

In reverse: factoring

You can also run the property backward. In $6x + 9$ , both terms have a factor of 3. Pull it out: $6x + 9 = 3(2x + 3)$ . This is called factoring, and you’ll use it a lot later in algebra.

Solving equations

Equations with parentheses often start with this step. In $2(x + 3) = 14$ , distribute first to get $2x + 6 = 14$ . Then solve it like the two-step equations you’ll see soon.

Practice Problems

Use the distributive property to simplify each expression.

$7(x + 2)$
$4(3y - 5)$
$-3(a + 6)$
$-5(2 - m)$
$(x - 9) \cdot 6$
$\frac{1}{2}(6x - 10)$
$-(4k - 7)$
$2(x + 3) + 5(x - 1)$
$10 - 4(y + 2)$
A rectangle is 4 meters wide and $2x + 3$ meters long. Write its area without parentheses. Then find the area when $x = 5$ .

Answers

Show answers

7 \cdot x + 7 \cdot 2

Multiply 7 by each term.

7x + 14

Multiply.

4(3y) + 4(-5)

Multiply 4 by each term.

12y - 20

Multiply.

(-3)(a) + (-3)(6)

Multiply −3 by each term.

-3a - 18

Multiply.

(-5)(2) + (-5)(-m)

Multiply −5 by each term. The second term is −m.

-10 + 5m

Multiply. A negative times a negative is positive.

5m - 10

Reorder, if you like.

x \cdot 6 + (-9) \cdot 6

Multiply each term by 6.

6x - 54

Multiply.

\frac{1}{2}(6x) + \frac{1}{2}(-10)

Multiply $\frac{1}{2}$ by each term (see multiplying and dividing fractions)

3x - 5

Multiply.

(-1)(4k) + (-1)(-7)

The minus sign means −1. Multiply −1 by each term.

-4k + 7

Multiply.

2x + 6 + 5(x - 1)

Distribute the 2.

2x + 6 + 5x - 5

Distribute the 5.

2x + 5x + 6 - 5

Move like terms together.

7x + 6 - 5

Combine the x-terms.

7x + 1

Combine the constants.

10 - 4y - 8

Distribute the −4.

-4y + 10 - 8

Move like terms together.

-4y + 2

Combine the constants.

10.

4(2x + 3)

Area is width times length.

8x + 12

Distribute the 4.

8(5) + 12

Replace x with 5.

40 + 12

Multiply.

52

Add.

Check: $4(2(5) + 3) = 4(13) = 52$
The area is 52 square meters.

Want more practice? The Algebra 1 Unit 1 Practice Pack has 26 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

Distributive Property FAQ

What is the distributive property?

It is the rule $a(b + c) = ab + ac$ . A number multiplied by a sum equals the sum of the number times each part.

What is an example of the distributive property?

$4(x + 5) = 4x + 20$ . With numbers only, $3(10 + 2) = 30 + 6 = 36$ , which is the same as $3(12)$ .

How do you use the distributive property with a negative number?

Multiply the negative number by every term and use the sign rules. For example, $-3(x - 4) = -3x + 12$ .

What is the distributive property used for?

It removes parentheses so you can combine like terms and solve equations. It also helps with mental math and with factoring.

Summary

The distributive property is $a(b + c) = ab + ac$ .
Multiply the number outside by every term inside the parentheses.
Watch the signs: a negative outside changes the sign of each term inside.
A minus sign in front of parentheses means multiply by $-1$ .
After distributing, combine like terms.

Previous lesson: Combining Like Terms

Next lesson: One-Step Equations

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