Introduction

In this lesson, you will learn about the distributive property of multiplication.

A property is a characteristic about something that is always true. The distributive property of multiplication is a rule that is always true. There are many properties in math that you use regularly, like the commutative property (the property that says that you can multiply in any order, \(2 \times 3 = 3 \times 2\)). The distributive property says that you can distribute a number being multiplied into parentheses.

These videos illustrate the lesson material below. Watching the videos is optional.

• Distributive Property of Multiplication (03:11 mins) | Transcript
• More Examples of the Distributive Property of Multiplication (03:13 mins) | Transcript

The Distributive Property of Multiplication

Example 1
\(2(1+3)\)

According to the order of operations, you must complete the math within the parentheses first.

\begin{align*} &2(1+3) &\color{red}\small\text{Simplify using order of operations}\\\\ &2(4) &\color{red}\small\text{Solve within the parentheses first}\\\\ &8&\color{red}\small\text{Multiply or divide from left to right}\\\\ \end{align*}

The answer is 8.

Looking at this equation another way can help you better understand what is going on. Using the black and orange dots as examples, you see that you have a group of 1 and a group of 3, and then both groups are multiplied by 2.

Figure 1

Figure 2

The distributive property of multiplication says that if you are multiplying something that’s within parentheses, you can either multiply everything within parentheses individually, or you can multiply after you have completed the operations inside the parentheses. When you solved the problem previously, you added together a group of one and a group of three. You then multiplied that answer by 2. If you look more closely at the circles, you can see that you really have:

\begin{align*}2(1&+3) &\color{red}\small\text{Simplify using the distributive property}\\\\2(1) &+ 2(3) &\color{red}\small\text{Distribute the \(2\) inside the parentheses}\\\\2 &+ 6 &\color{red}\small\text{Multiply from left to right}\\\\&8 &\color{red}\small\text{ Add or subtract from left to right}\\\end{align*}

Notice that regardless of which way you solve the problem, the answer is 8.

Example 2
\(-3(-4 + 2)\)
First solve it using the order of operations.

\begin{align*}-3(-4& + 2) &\color{red}\small\text{Simplify using order of operations}\\\\ -3(&-2) &\color{red}\small\text{Solve within the parentheses}\\\\&6 &\color{red}\small\text{Multiply from left to right last}\\\end{align*}

The answer is 6.

Remember, when there is a negative sign in front of both of the numbers, the answer will be a positive number.

Now solve the same problem using the distributive property of multiplication. The distributive property says to take the multiplier and distribute it to everything within the parentheses:

\begin{align*} -3(-&4 +2) &\color{red}\small\text{Simplify using the distributive property}\\\\ -3(-4) &+ -3(2) &\color{red}\small\text{Distribute the \(-3\) inside the parentheses}\\\\12 &- 6 &\color{red}\small\text{Multiply from left to right}\\\\ &6 &\color{red}\small\text{Add or subtract from left to right}\\\end{align*}

Again, the answer is 6.

Example 3
These expressions are examples of left-hand multipliers. However, you could just as easily place the multiplier on the right-hand side of the parentheses. The same principles that you learned from left-handed multipliers apply to right-handed multipliers.

• Left-Hand Multiplier:

\begin{align*} 3(4 &+1) &\color{red}\small\text{Simplify using the distributive property}\\\\ 3(4) &+ 3(1) &\color{red}\small\text{Distribute the \(3\) inside the parentheses}\\\\12 &+3 &\color{red}\small\text{Multiply from left to right}\\\\ &15 &\color{red}\small\text{ Add or subtract from left to right}\\\end{align*}

• Right-Hand Multiplier:

\begin{align*} (4 &+1)3 &\color{red}\small\text{Simplify using the distributive property}\\\\ 4(3) &+ 1(3) &\color{red}\small\text{Distribute the \(3\) inside the parentheses}\\\\12 &+3 &\color{red}\small\text{Multiply from left to right}\\\\ &15 &\color{red}\small\text{Add or subtract from left to right }\\\end{align*}
Example 4
\((3 + (-4))5\)
This equation has a set of parentheses contained within other parentheses. You can solve it using the same distributive property steps.

\begin{align*} (3 + (&-4)) 5 &\color{red}\small\text{Simplify using the distributive property}\\\\ 3(5) &+ -4(5) &\color{red}\small\text{Distribute the \(5\) inside the parentheses}\\\\15 &-20 &\color{red}\small\text{Multiply from left to right}\\\\ &-5 &\color{red}\small\text{Add or subtract from left to right }\\\end{align*}

The answer is -5.

Things to Remember

• Using either the order of operations or the distributive property of multiplication will give you the correct answer.
• When you multiply two negative numbers together, the answer is a positive number.

Practice Problems

1. \(4 \lgroup 3 + 4 \rgroup = ?\) (
   Solution

   x

   Solution: \(28\)
   Details:
   Although this problem can be solved using the order of operations, practice solving with the distributive property.
   Start by distributing the 4 into each number inside the parentheses. Multiply \(4 × 3\) and \(4 × 4\).
   
   \(4\lgroup3 + 4\rgroup\)
   \(4\lgroup3\rgroup + 4\lgroup4\rgroup\)
   
   Solve \(4 × 3 = 12\) and \(4 × 4 = 16\).
   
   \(4\lgroup3\rgroup + 4\lgroup4\rgroup\)
   \(12+16\)
   
   Finally add \(12 + 16 = 28\).
   )
2. \(\lgroup 2 + 9 \rgroup 3 = ?\) (
   Solution

   x

   Solution:
   33
   )
3. \(-8 \lgroup -2 + 5 \rgroup = ?\) (
   Video Solution

   x

   Solution: \(-24\)
   Details:
   

   
   (Distributive Property of Multiplication #3 (01:40 mins) | Transcript)
   | Transcript)
4. \(\lgroup -2 + 3 \rgroup \lgroup -6 \rgroup = ?\) (
   Solution)

   x

   Solution: \(-6\)
   Details:
   Solve by using the distributive property. Multiply \(-6\) by each number inside the parentheses.
   \(\lgroup-2 + 3\rgroup\lgroup-6\rgroup\)
   
   Continue to multiply \(-2 × -6\) and \(3 × -6\).
   \(\lgroup-2\rgroup\lgroup-6\rgroup + \lgroup3\rgroup\lgroup-6\rgroup\)
   
   The products, or answers, are \(12\) and \(-18\).
   \(\lgroup12\rgroup + \lgroup-18\rgroup\)
   
   The final step is to add:
   \(\lgroup12\rgroup + \lgroup-18\rgroup\)
   
   The answer is \(- 6\).
5. \(\lgroup -9 - 4 \rgroup \lgroup -8 \rgroup = ?\) (Video Solution | Transcript)
6. \(-5 \lgroup -7 - \lgroup -6 \rgroup \rgroup = ?\) (
   Solution

   x

   Solution:
   5
   )
7. \(\lgroup -5 - \lgroup -9 \rgroup \rgroup 6 = ?\) (
   Solution

   x

   Solution:
   24
   )

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