Distributive Property

Learn how to multiply a number or variable by a sum or difference inside parentheses.

Intermediate25 minLesson
Printable worksheet

Definition

The distributive property states that multiplying a number by a sum is the same as multiplying the number by each addend and then adding the products.
a(b+c)=ab+aca(b + c) = ab + ac
This also works with subtraction:
a(b−c)=ab−aca(b - c) = ab - ac
Key idea: The number outside the parentheses gets "distributed" to each term inside.

Try it now

Which expression is equivalent to 2(3+5)2(3 + 5)?

Worked Examples

Simplify: 4(5+3)4(5 + 3)
1

Identify the factor outside

The number 44 is outside the parentheses → Factor: 44
2

Distribute to each term

4×54 \times 5 and 4×34 \times 3 → 20+1220 + 12
3

Add the products

20+12=3220 + 12 = 32

Common Mistakes

Only distributing to the first term: 3(x+4)=3x+43(x + 4) = 3x + 4

Why it's wrong: The factor must multiply EVERY term inside the parentheses, not just the first one.

Correct: 3(x+4)=3x+123(x + 4) = 3x + 12 (multiply 33 by both xx AND 44)

Forgetting to change signs when distributing a negative: −2(x+3)=−2x+3-2(x + 3) = -2x + 3

Why it's wrong: A negative factor changes the sign of EACH product.

Correct: −2(x+3)=−2x−6-2(x + 3) = -2x - 6 (negative times positive equals negative)

Double negative error: −4(x−2)=−4x−8-4(x - 2) = -4x - 8

Why it's wrong: Negative times negative equals positive, not negative.

Correct: −4(x−2)=−4x+8-4(x - 2) = -4x + 8 (because −4×(−2)=+8-4 \times (-2) = +8)

Interactive Visual

Distributive property with numbers

a rows of (b + c) squares split into a rows of b and a rows of c: the total stays the same.

4×(5+3)4 \times (5 + 3)

4×(5+3)=4×5+4×34 \times (5 + 3) = 4 \times 5 + 4 \times 3

=20+12=32= 20 + 12 = 32

Area model: distributive property

a rows of (x + b): the area is a·x plus a·b.

3x3x
2121
xx
77
33

Length × width = area

3(x+7)=3x+213(x + 7) = 3x + 21

Interactive Sandbox

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which expression is equivalent to 2(3+5)2(3 + 5)?

Why It Matters

The distributive property is one of the most important tools in algebra:
  • Simplifying expressions: Turn 3(x+4)3(x + 4) into 3x+123x + 12
  • Mental math: Calculate 7×987 \times 98 as 7×100−7×2=700−14=6867 \times 100 - 7 \times 2 = 700 - 14 = 686
  • Factoring: Reverse the process to factor expressions
  • Real-world applications: Calculate totals when buying multiple items at different prices
Without the distributive property, many algebraic manipulations would be impossible!

Real World Applications

Shopping Calculations

The distributive property helps calculate totals when buying multiples of items with different components.

Example:

You buy 3 meal deals that each include a burger for 5 euros and fries for 2 euros. Total: 3(5+2)=3×5+3×2=15+6=213(5 + 2) = 3 \times 5 + 3 \times 2 = 15 + 6 = 21 euros.

1Try It Yourself

You are buying 4 gift bags. Each bag contains a toy for 8 euros and candy for 3 euros.

What is the total cost?

Step 1: Write the mathematical expression

Write the expression using distributive property:

Mental Math Tricks

Use the distributive property to make difficult calculations easier in your head.

Example:

To calculate 6×526 \times 52, think of it as 6(50+2)=300+12=3126(50 + 2) = 300 + 12 = 312.

2Try It Yourself

You need to calculate 8×478 \times 47 in your head.

How can you break this down?

Step 1: Write the mathematical expression

Rewrite as 8(50−?)8(50 - ?):

Key Takeaways

  • 1The distributive property: a(b+c)=ab+aca(b + c) = ab + ac
  • 2Multiply the outside factor by EACH term inside the parentheses
  • 3Works with both addition and subtraction
  • 4When distributing a negative, all signs change
  • 5Useful for simplifying expressions and mental math

Frequently Asked Questions

Because the factor outside the parentheses gets 'distributed' (shared) to each term inside. It's like distributing cookies equally to everyone!
Because the factor outside the parentheses gets 'distributed' (shared) to each term inside. It's like distributing cookies equally to everyone!
Yes, they're the same! Multiplication is commutative, so a(b+c)=(b+c)a=ab+aca(b + c) = (b + c)a = ab + ac.
Absolutely! For example, 2(x+y+z)=2x+2y+2z2(x + y + z) = 2x + 2y + 2z. Distribute to every term.

Glossary

Distributive property
A rule that states a(b+c)=ab+aca(b + c) = ab + ac
Factor
A number or expression that multiplies another (the number outside the parentheses)
Term
A part of an expression separated by + or - signs
Expand
To remove parentheses by applying the distributive property
Distribute
To multiply the outside factor by each term inside the parentheses

Formula Card

Distributive Property (Addition)

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

Multiply the factor by each term inside, then add

Distributive Property (Subtraction)

a(b−c)=ab−aca(b - c) = ab - ac

Multiply the factor by each term inside, keep the minus

Right-side Distribution

(b+c)a=ba+ca(b + c)a = ba + ca

The factor can also be on the right side

Today's challengeOne problem, three tries

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