Using 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.

Sample Set A

Using the order of operations, we get

\( \begin{aligned} 4(6+2) =4 \cdot 8 \\ =32 \end{aligned} \)

Using the order of operations, we get

\( \begin{aligned} 8(9+6) &=8 \cdot 15 \\ &=120 \end{aligned} \)

Source: Rice University, https://cnx.org/contents/XeVIW7Iw@4.6:Gmai8dss@2/Mental-Arithmetic-Using-the-Distributive-Property
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