Using the Distributive Property

Read this text, which explains why the distributive property works. Pay close attention to the worked examples in Sample Set A, which show how to use the distributive property. Do the examples in Practice Set A. Complete the practice questions and check your answers.

The Distributive Property

The distributive property is a characteristic of numbers that involves both addition and multiplication. It is used often in algebra, and we can use it now to obtain exact results for a multiplication.

Suppose we wish to compute $3(2+5)$ . We can proceed in either of two ways, one way which is known to us already (the order of operations), and a new way (the distributive property).

Compute $3(2+5)$ using the order of operations.

$3(2+5)$

Operate inside the parentheses first: $2+5 = 7$ .

$3(2+5) = 3⋅7$

Now multiply 3 and 7.

$3(2+5) = 3⋅7 = 21$

Thus $3(2+5) = 21$ .

Compute $3(2+5)$ using the distributive property.

We know that multiplication describes repeated addition. Thus,

$\begin{aligned} 3(2+5) &=\underbrace{2+5+2+5+2+5}_{2+5 \text { appears } 3 \text { times }} & & \\ &=2+2+2+5+5+5 & & \text { (by the commutative property of addition) } \\ &=3 \cdot 2+3 \cdot 5 & & \text { (since multiplication describes repeated addition) } \\ &=6+15 & & \\ &=21 & & \end{aligned}$

Thus, $3(2+5) = 21$ .

Let's look again at this use of the distributive property.

$\begin{array}{l} 3(2+5)=\underbrace{2+5+2+5+2+5}_{2+5 \text { appears } 3 \text { times }}\\ 3(2+5)=\underbrace{2+2+2}_{2 \text { appears } 3 \text { times }}+\underbrace{5+5+5}_{5 \text { appears } 3 \text { times }}\\ \begin{array}{l} 3(2+5)=3 \cdot 2+3 \cdot 5 \\ \quad \qquad \qquad 3 \times 2 \quad 3 \times 5 \end{array} \end{array}$

The 3 has been distributed to the 2 and 5.

This is the distributive property. We distribute the factor to each addend in the parentheses. The distributive property works for both sums and differences.