Math Core

How much do $6$ tickets cost at $98 each? You could multiply the long way, or think " $6$ tickets at $100 is $600, minus $2 back on each ticket is $12, so $588." That shortcut is the distributive property, and in algebra it's the tool for getting rid of parentheses.

The rule

The distributive property

For any numbers $a$ , $b$ and $c$ :

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

Multiply the number outside the parentheses by every term inside.

An area model shows why. A rectangle that is $a$ tall and $b + c$ wide can be split into two smaller rectangles, one $a$ by $b$ and one $a$ by $c$ . The total area is the same either way: $a(b + c) = ab + ac$ .

Rewriting $a(b + c)$ as $ab + ac$ is called expanding. For the tickets above, $6(98) = 6(100 - 2) = 600 - 12 = 588$ .

Worked example: Expanding

Expand $4(3x - 5)$ .

Multiply $4$ by each term inside: $4 \cdot 3x = 12x$ and $4 \cdot (-5) = -20$ .

4(3x - 5) = 12x - 20

Distributing a negative

When the number outside is negative, its sign multiplies every term inside, so every sign changes. A minus sign alone in front of parentheses means $-1$ : $-(x - 3) = -1 \cdot x + (-1)(-3) = -x + 3$ .

Worked example: A negative factor

Expand $-5(2y - 3)$ .

\begin{aligned} -5(2y - 3) &= (-5)(2y) + (-5)(-3) \\ &= -10y + 15 \end{aligned}

A negative times a negative is positive, so the last term is $+15$ .

Common mistake

The most common mistake is multiplying only the first term: $3(x + 4) \ne 3x + 4$ . The $3$ multiplies the $4$ too, so $3(x + 4) = 3x + 12$ .

With a negative factor, the second most common mistake is keeping a sign that should flip: $-2(n - 6) = -2n + 12$ , not $-2n - 12$ .

Distribute, then combine

Many expressions need both skills. Distribute first to clear the parentheses, then combine like terms.

Worked example: Two sets of parentheses

Simplify $3(2k + 1) - 2(k - 4)$ .

Think of the second part as distributing $-2$ , not $2$ .

\begin{aligned} 3(2k + 1) - 2(k - 4) &= 6k + 3 - 2k + 8 && \text{distribute } 3 \text{ and } -2 \\ &= (6k - 2k) + (3 + 8) && \text{group like terms} \\ &= 4k + 11 \end{aligned}

Tip

Check by substituting. With $k = 1$ : $3(3) - 2(-3) = 9 + 6 = 15$ , and $4(1) + 11 = 15$ . They match.

Factoring: the distributive property in reverse

You can also run the property backward. Factoring rewrites a sum as a product by pulling out a common factor. To factor completely, pull out the greatest common factor (GCF) of the terms.

Worked example: Factoring out the GCF

Factor $12x + 18$ .

The GCF of $12$ and $18$ is $6$ . Divide each term by $6$ : $12x \div 6 = 2x$ and $18 \div 6 = 3$ .

12x + 18 = 6(2x + 3)

Check by expanding: $6(2x + 3) = 12x + 18$ . Note that $2(6x + 9)$ is also equal, but it isn't completely factored because $6x + 9$ still has a common factor of $3$ .

Practice

Practice 1

Use the distributive property to find $7 \cdot 103$ in your head.

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

Practice 2

Expand $5(2x - 3)$ .

Practice 3

Expand $-4(3y - 2)$ .

Practice 4

Expand $-(5n - 9)$ . What is the constant term of the result?

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

Practice 5

Simplify $3(2k + 5) - 4k$ .

Practice 6

Simplify $4(x - 2) - 3(x - 5)$ . What is the constant term of the result?

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

Practice 7

Factor $18m + 24$ completely.

Practice 8

Simplify $2(3a - 1) - (a - 6) + 5a$ . What is the coefficient of $a$ in the result?

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