Distributive Property: Examples, Signs and Equations

Use the distributive property to multiply a factor by every term in parentheses. For example, \(3(x+4)=3x+12\). Keep each term’s sign, then combine like terms. Use the examples below to compare expanding an expression with solving an equation.

What Is the Distributive Property?

The distributive property states that multiplying a number by a sum (or difference) is the same as multiplying that number by each addend and then adding (or subtracting) the results:

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\(a(b + c) = \text{ ab } + \text{ ac }\)

\(a(b – c) = \text{ ab } – \text{ ac }\)

This works for any real numbers a, b, and c, and it works with variables too.

How to Apply the Distributive Property

With Numbers

Multiply the outside factor by each term inside the parentheses.

• \(3(2 + 4) = 3 \times 2 + 3 \times 4 = 6 + 12 = 18\) ✓ (same as \(3 \times 6 = 18\))
• \(5(6 – 3) = 5 \times 6 – 5 \times 3 = 30 – 15 = 15\) ✓

With Variables

The process is identical — multiply the outside term by every term inside.

• \(3(x + 4) = 3x + 12\)
• \(5(2x – 3) = 10x – 15\)

With a Negative Factor

A negative sign outside the parentheses flips the sign of every term inside.

• \(-2(x + 6) = -2x – 12\)
• \(-(x + 9) = -x – 9\)

Step-by-Step Summary

1. Identify the factor being multiplied outside the parentheses.
2. Multiply that factor by the first term inside the parentheses.
3. Multiply that factor by the second term (and every additional term).
4. Keep the same operation (+ or −) between the resulting terms.
5. Combine like terms if possible.

Watch: The Distributive Property (Algebra Basics)

Math Antics introduces the distributive property in its general algebraic form with clear examples:

The Distributive Property – Worked Examples

Example 1: Expand \(4(x + 5)\).

Multiply 4 by each term: \(4 \times x = 4x\) and \(4 \times 5 = 20\).
Result: \(4x + 20\). Check (\(x = 2\)): \(4(7) = 28 = 4(2)+20 = 28\) ✓

Example 2: Expand \(5(2x – 3)\).

Multiply 5 by each term: \(5 \times 2x = 10x\) and \(5 \times (-3) = -15\).
Result: \(10x – 15\).

Example 3: Expand \(-2(x + 6)\).

\(\text{ Multiply } -2\) by each term: \((-2) \times x = -2x\) and \((-2) \times 6 = -12\).
Result: \(-2x – 12\).

Example 4: Simplify \(3(x + 2) + 4(x – 1)\).

Distribute: \(3x + 6 + 4x – 4\). Combine like terms: \(7x + 2\).

More Practice: Distributive Property with Algebraic Expressions

Khan Academy applies the distributive property to algebraic expressions with additional examples:

Exercises for the Distributive Property

Expand each expression using the distributive property.

1. \(4(x + 5)\)
2. \(-3(2x – 1)\)
3. \(6(3 + x)\)
4. \(2(5x – 4)\)
5. \(-(x + 9)\)
6. \(3(x + 2) + 4(x – 1)\)

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Answers

1. \(4x + 20\)
2. \(-6x + 3\)
3. \(18 + 6x\)
4. \(10x – 8\)
5. \(-x – 9\)
6. \(7x + 2\)

Free The Distributive Property Worksheet

Ready to practice on your own? Download our free The Distributive Property worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again, steady practice is the surest way to master The Distributive Property before a quiz or test.

Download Variables Expressions and Properties Worksheet

Frequently Asked Questions

Does the distributive property work with subtraction?

Yes. \(a(b – c) = \text{ ab } – \text{ ac }\). Treat subtraction as adding a negative: \(a(b + (-c)) = \text{ ab } – \text{ ac }\).

Can I use the distributive property in reverse (factoring)?

Yes, that is called factoring out the GCF. For example, \(6x + 12 = 6(x + 2)\) because 6 is common to both terms.

Why does distributing a negative sign flip all the signs?

Because you are multiplying every term inside \(\text{ by } -1\). Multiplying \(\text{ by } -1\) reverses the sign: \((-1) \times a = -a\) and \((-1) \times (-a) = a\).

Related Topics

• Order of Operations
• Translate Phrases into an Algebraic Statement
• Multiplying and Dividing Integers
• Integers and Absolute Value
• Exponential Growth and Decay

For connected lessons, use the Algebra 1 learning center. For another explanation of the rule, see the OpenStax distributive property lesson.