Math Core

The distributive property is a two-way street. Going one way, you expand $4(x + 3)$ into $4x + 12$ . Going the other way, you factor $4x + 12$ back into $4(x + 3)$ . Both directions show up constantly in algebra, so it pays to get comfortable with each.

Expanding: multiply every term inside

Picture a rectangle that is $4$ units tall and $x + 3$ units wide. You can find its area in two ways:

All at once: height times width is $4(x + 3)$ .
In two pieces: a $4$ -by- $x$ part and a $4$ -by- $3$ part, for a total of $4x + 12$ .

It's the same rectangle, so $4(x + 3) = 4x + 12$ .

The area 4(x + 3) splits into 4x + 12.

The distributive property

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

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

Reading left to right is expanding. Reading right to left is factoring.

Worked example: Expanding with negatives and fractions

Expand each expression.

$-5(2x - 3)$
$\dfrac{2}{3}(6y + 9)$

Solutions.

Multiply $-5$ by each term: $(-5)(2x) + (-5)(-3) = -10x + 15$ . A negative times a negative is positive.
Multiply $\dfrac{2}{3}$ by each term: $\dfrac{2}{3} \cdot 6y + \dfrac{2}{3} \cdot 9 = 4y + 6$ .

Common mistake

The most common mistake is multiplying only the first term. $-5(2x - 3)$ is not $-10x - 3$ . The outside number multiplies every term inside, and a negative outside flips every sign.

A minus sign in front of parentheses means "multiply by $-1$ ." So $-(4 - k) = -4 + k$ .

Factoring: undo the distributing

To factor an expression like $6x + 15$ , look for a number that divides every term. The biggest such number is the greatest common factor (GCF).

Definition

Factoring

Factoring means rewriting a sum as a product. To factor out a common factor, write it in front of parentheses, and inside write what's left of each term after dividing by it.

For $6x + 15$ : the GCF of $6$ and $15$ is $3$ . Divide each term by $3$ : $6x \div 3 = 2x$ and $15 \div 3 = 5$ . So

6x + 15 = 3(2x + 5).

Worked example: Factoring out the GCF

Factor $12n - 20$ completely.

The GCF of $12$ and $20$ is $4$ . Divide each term: $12n \div 4 = 3n$ and $-20 \div 4 = -5$ .

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

Check by expanding: $4(3n) - 4(5) = 12n - 20$ . It matches.

Notice that $2(6n - 10)$ is also equal to $12n - 20$ , but it isn't factored completely, because $6n - 10$ still has a common factor of $2$ .

Tip

Always check a factored answer by expanding it. You should get back exactly the expression you started with.

Factoring out a negative

Sometimes you want to pull out a negative number instead of the positive GCF, for example when the first term is negative. Just divide each term by it, being careful with signs.

Worked example: Factoring out a negative

Rewrite $-8x + 24$ as $-8$ times a sum.

Divide each term by $-8$ : $-8x \div (-8) = x$ and $24 \div (-8) = -3$ .

-8x + 24 = -8(x - 3)

Check: $(-8)(x) + (-8)(-3) = -8x + 24$ . It matches.

Worked example: A real-world factor

Maya buys $5$ packs of stickers. Each pack costs $p$ dollars, and she pays a $2 shipping fee per pack. Write her total cost two ways.

Per pack: each pack costs $p + 2$ dollars, so $5$ packs cost $5(p + 2)$ .
By type: the stickers cost $5p$ dollars and shipping costs $5 \cdot 2 = 10$ dollars, for $5p + 10$ .

The factored form $5(p + 2)$ shows the cost of one pack. The expanded form $5p + 10$ shows the total shipping, $10.

Practice

Practice 1

Which expression is equal to $7(x + 4)$ ?

Practice 2

When $-6(2a - 3)$ is expanded, what is the constant term?

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

Practice 3

Which expression is equal to $\dfrac{3}{4}(8m - 12)$ ?

Practice 4

What is the greatest common factor of $18x$ and $30$ ?

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

Practice 5

Which shows $14y + 35$ factored completely?

Practice 6

Fill in the blank: $-10k + 50 = -10(k - \underline{\quad})$ .

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

Practice 7

Which expression is not equal to $24x + 16$ ?

Practice 8

A class orders $6$ pizzas. Each pizza costs $c$ dollars plus a delivery charge of $d$ dollars per pizza. The total is $6c + 24$ dollars. Factor to find the delivery charge per pizza, $d$ .

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