5.2 The Distributive Property

Simplify Expressions Using the Distributive Property

Suppose three friends are going to the movies. They each need $\text{\$9.25};$ that is, $9$ dollars and $1$ quarter. How much money do they need all together? You can think about the dollars separately from the quarters.

They need $3$ times $\text{\$9},$ so $\text{\$27},$ and $3$ times $1$ quarter, so $75$ cents. In total, they need $\text{\$27.75}.$

If you think about doing the math in this way, you are using the Distributive Property.

Back to our friends at the movies, we could show the math steps we take to find the total amount of money they need like this:

$$\begin{array}{ccc}& \hfill 3(9.25)& \\ \hfill 3(9& +& 0.25)\hfill \\ \hfill 3(9)& +& 3(0.25)\hfill \\ \hfill 27& +& 0.75\hfill \\ & \hfill 27.75& \end{array}$$

In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression $3(x+4),$ the order of operations says to work in the parentheses first. But we cannot add $x$ and $4,$ since they are not like terms. So we use the Distributive Property, as shown in Example 1.

Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in Example 1 would look like this:

The distributive property can be used to simplify expressions that look slightly different from $a(b+c).$ Here are two other forms.

In the next two examples, we multiply a fraction by a whole number.

Using the Distributive Property as shown in the next example will be very useful when we clear decimals from equations.

In the next example we’ll multiply by a variable.

The next example will use the ‘backwards’ form of the Distributive Property, $(b+c)a=ba+ca.$

When you distribute a negative number, you need to be extra careful to get the signs correct.

In the next example, we will show how to use the Distributive Property to find the opposite of an expression. Remember, $-a=\mathrm{-1}·a.$

Sometimes we need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.

Practice Makes Perfect

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the distributive property.