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Distributive Property

What is the Distributive Property?

The distributive property is a fundamental property of mathematics that allows us to multiply a number by a sum or difference by distributing the multiplication over each term.

Distributive Property Formula

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

When a number multiplies a sum or difference, we can distribute the multiplication to each term inside the parentheses.

Examples of the Distributive Property

Example 1: Distributing with Positive Numbers

3(4 + 5)
= 3 × 4 + 3 × 5
= 12 + 15
= 27

We can verify this is correct: 3(4 + 5) = 3(9) = 27

Example 2: Distributing with Negative Numbers

-2(6 - 8)
= -2 × 6 - (-2) × 8
= -12 + 16
= 4

We can verify this is correct: -2(6 - 8) = -2(-2) = 4

Example 3: Distributing with Variables

5(x + 3)
= 5 × x + 5 × 3
= 5x + 15

This is a common application in algebra, where we distribute a coefficient to terms with variables.

Using the Distributive Property to Simplify Expressions

The distributive property is a powerful tool for simplifying algebraic expressions and solving equations.

Simplifying Expressions

7(2x + 4) - 3(x - 1)
= 14x + 28 - 3x + 3
= 14x - 3x + 28 + 3
= 11x + 31

Factoring Using the Distributive Property

We can also use the distributive property in reverse to factor expressions.

6x + 15
= 3(2x) + 3(5)
= 3(2x + 5)

We found the greatest common factor (GCF) of 6x and 15, which is 3, and factored it out.

Real-World Applications

The distributive property has many practical applications in everyday life.

Shopping Example

Imagine you're buying 3 shirts that cost $15 each and 2 pairs of socks that cost $5 each. You can calculate the total cost using the distributive property.

Total cost = 3($15) + 2($5)
= $45 + $10
= $55

Mental Math

The distributive property can help with mental calculations. For example, to multiply 7 × 98, you can rewrite it as:

7 × 98 = 7 × (100 - 2)
= 7 × 100 - 7 × 2
= 700 - 14
= 686

This is often easier than multiplying 7 and 98 directly.

Related Topics

Practice Distributive Property

Test your knowledge with our interactive distributive property practice exercises.

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Equation Solver

Use our calculator to solve equations step by step.

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Key Concepts

  • 1
    The distributive property states that a(b + c) = ab + ac.
  • 2
    It works with addition and subtraction: a(b - c) = ab - ac.
  • 3
    The distributive property can be used to simplify algebraic expressions.
  • 4
    It can also be used in reverse for factoring expressions.
  • 5
    The distributive property has many practical applications in everyday life and mental math.