Lesson 21: Distributive Property and Order of Operations

New Concept

This property lets you multiply a sum by multiplying each addend separately before adding. It is essential for expanding and factoring algebraic expressions.

What’s next

You’ll start by using this property to expand and factor expressions. Then, you’ll see how it fits within the order of operations to solve multi-step problems.

Property

To expand an expression, multiply the term outside the parentheses by each term inside: $a(b+c) = a \cdot b + a \cdot c$.

Examples

• $3(w+m) = 3w+3m$
• $5(x-3) = 5x-15$
• $2(x+7) = 2x+14$

Explanation

Think of it as sharing snacks! The number outside the parentheses, 'a', has to be distributed to every single friend, 'b' and 'c', inside. No one gets left out of the multiplication party, ensuring everyone gets their share of the mathematical treat!

Property

To factor an expression, find the greatest common factor (GCF) and pull it out, reversing the distribution: $ab+ac = a(b+c)$.

Examples

• $ax+ay = a(x+y)$
• $6x+9 = 3(2x+3)$
• $8w-10 = 2(4w-5)$

Explanation

Factoring is like being a detective! You must examine each term to find the greatest common factor they all share. Once you find it, you pull it out to the front of the parentheses, revealing the simplified expression hidden inside. It's reverse distribution!

Let’s flip the script: can you spot what’s shared and, like a math magician, tuck it outside?

Example Problem:
Factor: $8x + 12$

Step-by-Step:

1. Look for a number that divides evenly into both $8x$ and $12$. That’s our common factor.
2. $8x$ can be written as $4 \times 2x$ and $12$ as $4 \times 3$.
3. Using the distributive property in reverse (factoring), write: $4(2x) + 4(3) = 4(2x + 3)$.
4. So, $8x + 12 = 4(2x + 3)$.