Distributive Property Examples and Practice

What Is the Distributive Property?

The distributive property is a math rule that lets you simplify expressions with parentheses. It states that multiplying a number by a sum equals the sum of that number multiplied by each addend separately.

Sounds complicated. It's not.

Here's the formula in its simplest form:

a(b + c) = ab + ac

That's it. You take the number outside the parentheses and distribute it to each term inside. Break it down, solve each part, then add or subtract the results.

Distributive Property Examples

Basic Example

3(4 + 5) = ?

Without the property: 3(9) = 27

With the property: 3(4) + 3(5) = 12 + 15 = 27

Same answer. The property just shows you the intermediate steps.

Example with Variables

2(x + 7)

Distribute: 2(x) + 2(7) = 2x + 14

The 2 multiplies both x and 7. That's all that's happening.

Example with Subtraction

5(y - 3)

Distribute: 5(y) - 5(3) = 5y - 15

The subtraction sign stays. You distribute the multiplication across both terms, keeping the operations intact.

Example with Negative Numbers

-2(a + 6)

Distribute: -2(a) + (-2)(6) = -2a - 12

Negative times positive gives negative. Don't forget the negative sign carries through.

Example with Multiple Variables

4(x + y)

Distribute: 4x + 4y

Variables stay separate. You can't combine them unless you have more information.

Distributive Property with Exponents

Things get trickier when variables have exponents.

3x²(x + 4)

Distribute: 3x²(x) + 3x²(4) = 3x³ + 12x²

When multiplying powers with the same base, add the exponents: x² · x = x³.

Common Mistake to Avoid

Wrong: 2x(x + 3) = 2x² + 3

Right: 2x(x + 3) = 2x² + 6x

The 2x multiplies both terms. The 3 gets multiplied too, not ignored.

Distributive Property with Coefficients

7(2x + 3)

Step 1: 7 × 2x = 14x

Step 2: 7 × 3 = 21

Step 3: 14x + 21

Each term inside the parentheses gets multiplied by 7. No exceptions.

How to Use the Distributive Property: Step-by-Step

Here's the process for any expression:

1. Identify the term outside the parentheses. This is your multiplier.
2. Multiply it by the first term inside. Keep track of signs (+ or -).
3. Multiply it by the second term inside. And every term after that.
4. Write the results as a sum or difference. Combine like terms if possible.

Practice Problem

Solve: 6(2x + 5) - 3(x - 4)

Step 1: 6(2x + 5) = 12x + 30

Step 2: -3(x - 4) = -3x + 12

Step 3: Combine: 12x + 30 - 3x + 12

Step 4: Simplify: 9x + 42

Distributive Property vs. FOIL Method

Students confuse these constantly. They're not the same thing.

FOIL is actually just the distributive property applied twice. You're distributing (x + 2) across (x + 3), then distributing x and 2 separately. The distributive property is the foundation.

Distributive Property Practice Problems

Try these. Answers below.

1. 4(x + 7)
2. 5(2y - 3)
3. -3(4a + 2)
4. 2x(x + 6)
5. 7(3m - 5) + 2(4m + 1)

Answers

1. 4x + 28
2. 10y - 15
3. -12a - 6
4. 2x² + 12x
5. 21m - 35 + 8m + 2 = 29m - 33

Where the Distributive Property Shows Up

You need this for:

• Simplifying algebraic expressions — combining like terms after distribution
• Solving equations — isolating variables by distributing and rearranging
• Factoring — reversing the process to pull out common terms
• Polynomial multiplication — handling products with multiple terms
• Mental math shortcuts — breaking down 6 × 14 as 6(10 + 4)

The Bottom Line

The distributive property is just breaking down multiplication into smaller pieces. Take the number outside, multiply it by each term inside, then add or subtract the results.

Master this and algebra gets significantly easier. It's not optional—it's the backbone of half the problems you'll encounter.