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Mom's Basement University

The Distributive Property

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn the distributive property and how to use it to expand or simplify expressions.

The Concept

Section titled “The Concept”

The distributive property says you can multiply a number by each term inside parentheses:

a(b+c)=ab+aca(b + c) = ab + ac

Also works for subtraction and negatives:

3(4−2x)=3(4)−3(2x)=12−6x3(4 - 2x) = 3(4) - 3(2x) = 12 - 6x
  • Multiply the outside number by every term inside.
  • Keep the signs (+ or −) as they are.
  • Don’t forget to distribute to every term.

Example: 5(2x+3)=10x+155(2x + 3) = 10x + 15

Worked Examples

Section titled “Worked Examples”

Expand 4(3y−5)4(3y - 5)

  1. 4×3y=12y4 \times 3y = 12y
  2. 4×(−5)=−204 \times (-5) = -20
  3. Result: 12y−2012y - 20

Now with negative: −2(x+7)-2(x + 7)

  1. −2×x=−2x-2 \times x = -2x
  2. −2×7=−14-2 \times 7 = -14
  3. −2x−14-2x - 14

Real-World Applications

Section titled “Real-World Applications”

The distributive property helps break down costs or totals. Example: A plan costs 3 dollars per day for 5 days plus a 10-dollar fee: 3(5)+10=15+10=253(5) + 10 = 15 + 10 = 25 dollars. Or splitting a 120-dollar bill among 4 people with 5 dollars extra each: 4(30+5)=4×30+4×5=120+20=1404(30 + 5) = 4 \times 30 + 4 \times 5 = 120 + 20 = 140 dollars total.

Quiz

Section titled “Quiz”
Use distributive property: $6(2x + 3)$.
Expand $5(4 - y)$.
A fee is 8 dollars per person for 3 people plus 12 dollars flat. Simplified?
Distribute $-3(x - 2)$.
Expand and simplify $4(2x + 1) + 3$.