Math Core

The order of operations says to work inside parentheses first. But what if you can't, as in $3(x + 4)$ , where $x + 4$ won't simplify any further? The distributive property gives you another way in: multiply the outside number by each term inside.

Why it works

Suppose you buy $3$ bags, and each bag holds $x$ apples and $4$ oranges. How many pieces of fruit do you have?

Count bag by bag: each bag has $x + 4$ pieces, so the total is $3(x + 4)$ .
Count by type: there are $3x$ apples and $3 \cdot 4 = 12$ oranges, so the total is $3x + 12$ .

Both counts describe the same fruit, so $3(x + 4) = 3x + 12$ . The $3$ is "distributed" to each term in the parentheses.

You can also see it as area. A rectangle $3$ units tall and $x + 4$ units wide splits into two smaller rectangles, $3$ by $x$ and $3$ by $4$ . The whole area $3(x + 4)$ equals the sum of the parts, $3x + 12$ .

The distributive property

For all real numbers $a$ , $b$ and $c$ :

a(b + c) = ab + ac \qquad\qquad a(b - c) = ab - ac

It also works from the right: $(b + c)a = ba + ca$ , and with more than two terms inside: $a(b + c + d) = ab + ac + ad$ .

Worked example: Distributing a positive number

Rewrite without parentheses.

$5(x + 2)$
$4(3y - 7)$
$(2n + 1) \cdot 6$

Solutions.

$5 \cdot x + 5 \cdot 2 = 5x + 10$
$4 \cdot 3y - 4 \cdot 7 = 12y - 28$
$2n \cdot 6 + 1 \cdot 6 = 12n + 6$

Distributing a negative number

When the number outside is negative, it multiplies every term inside, and it changes each sign. The safest method is to think of subtraction as adding a negative term.

Worked example: A negative outside

Rewrite $-3(2x - 5)$ without parentheses.

Think of the inside as $2x + (-5)$ and multiply each term by $-3$ :

\begin{aligned} -3(2x - 5) &= (-3)(2x) + (-3)(-5) \\ &= -6x + 15 \end{aligned}

The last term is $+15$ because a negative times a negative is positive.

Common mistake

The most common mistake is multiplying only the first term: $-3(2x - 5) \ne -6x - 5$ . Draw an arrow from the outside number to each term inside, and give every arrow its own multiplication.

A lone minus sign in front of parentheses means "multiply by $-1$ ." So $-(x - 8) = -1(x - 8) = -x + 8$ . Every sign inside flips.

Worked example: Subtracting a group

Simplify $10 - (4 + y)$ .

The minus sign applies to the whole group, so rewrite it as $10 + (-1)(4 + y)$ :

10 - (4 + y) = 10 - 4 - y = 6 - y.

A common error is to write $10 - 4 + y$ , which changes the sign of only the $4$ .

The distributive property and mental math

Splitting a number into friendly pieces lets you multiply in your head.

Worked example: Multiplying quickly

Compute $7 \cdot 98$ and $6 \cdot 45$ without a calculator.

Write $98$ as $100 - 2$ :

7 \cdot 98 = 7(100 - 2) = 700 - 14 = 686.

Write $45$ as $40 + 5$ :

6 \cdot 45 = 6(40 + 5) = 240 + 30 = 270.

Tip

Check a distribution by substituting a number. If $x = 1$ , then $-3(2x - 5) = -3(-3) = 9$ , and $-6x + 15 = -6 + 15 = 9$ . The values match, so the rewrite is right. (If they don't match, you've made an error somewhere.)

Fractions and decimals

The property works the same way with any real numbers.

$\dfrac{1}{2}(8x + 6) = 4x + 3$
$0.5(4a - 10) = 2a - 5$
$\dfrac{2}{3}(9m - 3) = 6m - 2$

Practice

Practice 1

Which expression is equal to $6(x + 5)$ ?

Practice 2

Which expression is equal to $-4(3y - 2)$ ?

Practice 3

When $7(2x - 3)$ is rewritten without parentheses, what is the constant term?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Which expression is equal to $-(a - 9)$ ?

Practice 5

Use the distributive property to compute $5 \cdot 99$ in your head.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

When $\dfrac{2}{5}(15n - 25)$ is rewritten without parentheses, what is the constant term?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

Which expression is equal to $-2(x^2 - 4x + 3)$ ?

Practice 8

A rectangular garden is $8$ feet wide. Its length is $x + 5$ feet. Its area is $8(x + 5)$ square feet, which can be rewritten as $8x + 40$ . What is the area, in square feet, when $x = 7$ ? Compute it both ways to check.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.