Distributive Property

Simplify expressions using the distributive property

Suppose three friends are going to the movies. They each need $9.25 — that’s $9$ dollars and $1$ quarter. How much money do they need altogether? You can think about the dollars separately from the quarters: $3$ times $9 is $27, and $3$ times $1$ quarter is $75$ cents. In total, they need $27.75.

If you think about doing the math 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:

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 the next example.

Example. Simplify: $3(x + 4)$.

Distribute, then multiply:

Some students find it helpful to draw arrows to remind them how to use the Distributive Property — an arrow from the $6$ to each term inside the parentheses, so $6(5y + 1)$ becomes $6 \cdot 5y + 6 \cdot 1 = 30y + 6$.

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

Example. Simplify: $2(x - 3)$.

Distribute, then multiply:

Do you remember how to multiply a fraction by a whole number? We’ll need to do that in the next two examples.

Example. Simplify: $\tfrac{3}{4}(n + 12)$.

Distribute, then simplify:

Example. Simplify: $8\left(\tfrac{3}{8}x + \tfrac{1}{4}\right)$.

Distribute, then multiply:

Using the Distributive Property as shown in the next example will be very useful when we solve money applications later.

Example. Simplify: $100(0.3 + 0.25q)$.

Distribute, then multiply:

In the next example we’ll multiply by a variable. We’ll need to do this in a later chapter.

Example. Simplify: $m(n - 4)$.

Distribute, then multiply:

Notice that we wrote $m \cdot 4$ as $4m$. We can do this because of the Commutative Property of Multiplication — when a term is the product of a number and a variable, we write the number first.

The next example uses the “backwards” form of the Distributive Property, $(b + c)a = ba + ca$.

Example. Simplify: $(x + 8)p$.

Distribute:

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

Example. Simplify: $-2(4y + 1)$.

Distribute, then simplify:

Example. Simplify: $-11(4 - 3a)$.

Distribute, then simplify:

You could also write the result as $33a - 44$. Do you know why?

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

Example. Simplify: $-(y + 5)$.

Multiplying by $-1$ results in the opposite, then distribute:

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 cannot be simplified, the next step is to multiply using the Distributive Property, which removes the parentheses.

Example. Simplify: $8 - 2(x + 3)$.

Distribute, multiply, then combine like terms:

Example. Simplify: $4(x - 8) - (x + 3)$.

Distribute both terms, then combine like terms:

Evaluate expressions using the distributive property

Some students need to be convinced that the Distributive Property always works. In the examples below, we’ll practice evaluating some of the expressions from previous examples — in part (a), we’ll evaluate the form with parentheses, and in part (b) we’ll evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.

Example. When $y = 10$, evaluate: (a) $6(5y + 1)$ (b) $6 \cdot 5y + 6 \cdot 1$.

(a) Substituting $10$ for $y$ and simplifying in the parentheses first:

(b) Substituting $10$ for $y$ and simplifying:

Notice that the answers are the same: when $y = 10$, $6(5y + 1) = 6 \cdot 5y + 6 \cdot 1$.

Example. When $y = 3$, evaluate: (a) $-2(4y + 1)$ (b) $-2 \cdot 4y + (-2) \cdot 1$.

Both forms evaluate to $-26$, confirming that $-2(4y + 1) = -8y - 2$ when $y = 3$.

Example. When $y = 35$, evaluate (a) $-(y + 5)$ and (b) $-y - 5$, to show that $-(y + 5) = -y - 5$.

(a) Substituting $35$ for $y$: $-(35 + 5) = -(40) = -40$.

(b) Substituting $35$ for $y$: $-35 - 5 = -40$.

The answers are the same when $y = 35$, demonstrating that $-(y + 5) = -y - 5$.

Key terms

Distributive Property — for real numbers $a$, $b$, $c$: $a(b + c) = ab + ac$; also written $a(b - c) = ab - ac$ and $(b + c)a = ba + ca$. Multiplying a sum (or difference) by a number gives the same result as multiplying each term by that number and then adding (or subtracting) the products.

This section is adapted from Prealgebra 2e, Section 7.3: Distributive Property by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: described the opening bills-and-coins illustration in prose instead of recreating it as a graphic; omitted the Be Prepared quiz, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.