Distributive Property Formula

The Formula

a(b + c) = ab + ac

When to use: Three packs of (2 red + 4 blue) = (3 \times 2 red) + (3 \times 4 blue) = 6 red + 12 blue.

Quick Example

5(10 + 2) = 5 \times 10 + 5 \times 2 = 50 + 10 = 60 Easier than 5 \times 12 directly.

Notation

a(b + c) is shorthand for a \times (b + c); the factor distributes to each term inside

What This Formula Means

The rule that multiplying a sum equals the sum of individual products: a(b+c) = ab + ac. It links multiplication and addition, allowing you to break apart or combine terms.

Three packs of (2 red + 4 blue) = (3 \times 2 red) + (3 \times 4 blue) = 6 red + 12 blue.

Formal View

\forall a, b, c \in \mathbb{R}: a(b + c) = ab + ac \text{ and } (a + b)c = ac + bc

Worked Examples

Example 1

easy
Expand 4(x + 7).

Solution

  1. 1
    Identify the two terms inside the parentheses: x and 7.
  2. 2
    Distribute 4 to each term inside the parentheses: 4 \cdot x + 4 \cdot 7.
  3. 3
    Simplify: 4x + 28.

Answer

4x + 28
The distributive property states a(b + c) = ab + ac. Multiply the outside factor by each term inside the parentheses.

Example 2

medium
Expand and simplify 3(2x - 5) + 2(x + 4).

Common Mistakes

  • Only distributing to the first term inside the parentheses: 3(x + 4) = 3x + 4 instead of 3x + 12
  • Forgetting to distribute the sign: -(a + b) = -a + b instead of -a - b
  • Trying to distribute multiplication over multiplication: a(bc) \neq (ab)(ac)

Why This Formula Matters

Central to algebra, mental math shortcuts, and factoring. The distributive property powers techniques from expanding polynomials to simplifying complex expressions.

Frequently Asked Questions

What is the Distributive Property formula?

The rule that multiplying a sum equals the sum of individual products: a(b+c) = ab + ac. It links multiplication and addition, allowing you to break apart or combine terms.

How do you use the Distributive Property formula?

Three packs of (2 red + 4 blue) = (3 \times 2 red) + (3 \times 4 blue) = 6 red + 12 blue.

What do the symbols mean in the Distributive Property formula?

a(b + c) is shorthand for a \times (b + c); the factor distributes to each term inside

Why is the Distributive Property formula important in Math?

Central to algebra, mental math shortcuts, and factoring. The distributive property powers techniques from expanding polynomials to simplifying complex expressions.

What do students get wrong about Distributive Property?

Works forwards (distributing) and backwards (factoring): ab + ac = a(b+c) is equally valid.

What should I learn before the Distributive Property formula?

Before studying the Distributive Property formula, you should understand: multiplication, addition.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Commutative, Associative, and Distributive Properties →
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