The Distributive Property

What is the Distributive Property?

The distributive property says you can distribute (multiply) a number across terms inside parentheses.

Formula: a(b + c) = ab + ac

In words: Multiply the outside number by each term inside the parentheses.

Why It Works

Example: 3(4 + 2)

Method 1: Add first

• 3(4 + 2) = 3(6) = 18

Method 2: Distribute

• 3(4 + 2) = 3 × 4 + 3 × 2 = 12 + 6 = 18

Same answer! Both methods work.

Basic Examples

Example: 5(x + 3)

Distribute 5:

• 5 × x = 5x
• 5 × 3 = 15

Answer: 5x + 15

Example: 4(2 + y)

Distribute 4:

• 4 × 2 = 8
• 4 × y = 4y

Answer: 8 + 4y (or 4y + 8)

Example: 6(a + 7)

Distribute 6:

• 6 × a = 6a
• 6 × 7 = 42

Answer: 6a + 42

Distributing with Subtraction

Formula: a(b - c) = ab - ac

Example: 3(x - 4)

Distribute 3:

• 3 × x = 3x
• 3 × (-4) = -12

Answer: 3x - 12

Example: 7(5 - n)

Distribute 7:

• 7 × 5 = 35
• 7 × (-n) = -7n

Answer: 35 - 7n

Distributing Negative Numbers

Example: -2(x + 5)

Distribute -2:

• (-2) × x = -2x
• (-2) × 5 = -10

Answer: -2x - 10

Example: -3(4 - y)

Distribute -3:

• (-3) × 4 = -12
• (-3) × (-y) = 3y

Answer: -12 + 3y (or 3y - 12)

Important: Negative times negative = positive!

Distributing with Multiple Terms

Example: 2(3x + 4y - 5)

Distribute 2 to each term:

• 2 × 3x = 6x
• 2 × 4y = 8y
• 2 × (-5) = -10

Answer: 6x + 8y - 10

Distributing Fractions

Example: (1/2)(x + 6)

Distribute 1/2:

• (1/2) × x = (1/2)x or x/2
• (1/2) × 6 = 3

Answer: (1/2)x + 3 or x/2 + 3

Example: (2/3)(9 - 3x)

Distribute 2/3:

• (2/3) × 9 = 18/3 = 6
• (2/3) × (-3x) = -6x/3 = -2x

Answer: 6 - 2x

Working Backwards (Factoring)

The reverse of distributing is called factoring — pulling out a common factor.

Example: 4x + 12

Notice: Both terms have 4 as a factor

• 4x = 4 × x
• 12 = 4 × 3

Factor out 4:

• 4x + 12 = 4(x + 3)

Example: 15 - 5y

Common factor: 5

• 15 = 5 × 3
• 5y = 5 × y

Factor out 5:

• 15 - 5y = 5(3 - y)

Using Distributive Property to Multiply

Mental math trick: 7 × 98

Rewrite 98 as (100 - 2):

• 7 × 98 = 7(100 - 2)
• = 7 × 100 - 7 × 2
• = 700 - 14
• = 686

Example: 6 × 52

Rewrite as 6(50 + 2):

• = 6 × 50 + 6 × 2
• = 300 + 12
• = 312

Combining with Like Terms

After distributing, you can often simplify further.

Example: 3(x + 2) + 4x

Step 1: Distribute

• 3x + 6 + 4x

Step 2: Combine like terms

• 3x + 4x + 6
• 7x + 6

Answer: 7x + 6

Example: 5(2y - 1) - 3(y + 4)

Step 1: Distribute both

• 10y - 5 - 3y - 12

Step 2: Combine like terms

• 10y - 3y - 5 - 12
• 7y - 17

Answer: 7y - 17

Real-World Applications

Shopping: You buy 3 items for $x each plus $5 shipping per item

• Cost = 3(x + 5) = 3x + 15

Geometry: Perimeter of rectangle with length (x + 4) and width 2

• P = 2(x + 4) + 2(2) = 2x + 8 + 4 = 2x + 12

Practice