Introduction

The distributive property is one of the most useful tools in algebra. It allows us to expand expressions like $$3(x + 4)$$ or $$5(2y - 7)$$ by “distributing” the number outside the parentheses across each term inside.

This article assumes you are comfortable with basic arithmetic (addition, subtraction, multiplication).

What Is the Distributive Property?

The distributive property states: $$a(b + c) = ab + ac$$ Key ideas:

• The number outside the parentheses multiplies each term inside.
• Works with addition and subtraction.
• Helps simplify expressions and prepare them for solving equations.

Examples:

• $4(x + 2) = 4x + 8$
• $6(3 - y) = 18 - 6y$

Why the Distributive Property Matters

You will use this property constantly in algebra because it helps you:

• Remove parentheses.
• Combine like terms more easily.
• Rewrite expressions in simpler or more useful forms.
• Understand how multiplication interacts with addition and subtraction.

It also appears in:

• Factoring
• Solving equations
• Working with polynomials
• Geometry and area models

Expanding Expressions Step-by-Step

Here is a simple process you can follow:

1. Identify the multiplier

This is the number (or variable) outside the parentheses.

2. Multiply it by each term inside

• Multiply signs carefully.
• Multiply numbers and variables separately if needed.

3. Rewrite the expression without parentheses

• Keep the order clear.
• Combine like terms if possible.

Examples

• $5(x + 3)$
  → $5x + 15$
• $-2(4 - y)$
  → $-8 + 2y$
• $3(2x + 5y - 1)$
  → $6x + 15y - 3$

Common Mistakes to Avoid

• ❌ Forgetting to multiply every term inside the parentheses
• ❌ Dropping negative signs
• ❌ Mixing up addition and subtraction
• ❌ Thinking the distributive property works with multiplication inside parentheses (it does not): $$a(bc) \neq ab + ac$$

Exercises

1. Expand the expression: $4(x + 7)$

   Solution

   $$4(x + 7) = 4x + 28$$

2. Expand: $-3(2y - 5)$

   Solution

   $$-3(2y - 5) = -6y + 15$$

3. Expand and simplify: $2(3x + 4) + x$

   Solution

   $$2(3x + 4) + x = 6x + 8 + x = 7x + 8$$

4. Expand: $5(a - 2b + 3)$

   Solution

   $$5(a - 2b + 3) = 5a - 10b + 15$$

5. True or false: The distributive property can be used to expand $7(xy)$

   Solution

   False.
   $7(xy)$ is already a single multiplication; there is nothing to distribute.

6. Expand: $-2(4 - 3x)$

   Solution

   $$-2(4 - 3x) = -8 + 6x$$

7. Expand and simplify: $3(x - 1) + 2(x + 5)$

   Solution

   $$3(x - 1) + 2(x + 5)$$ $$= 3x - 3 + 2x + 10$$ $$= 5x + 7$$