Multiply Using the Distributive Property- Algebraic Techniques Explained
What the Distributive Property Actually Is
The distributive property is a simple rule that lets you break multiplication problems into smaller pieces. It states that a(b + c) = ab + ac. That's it. That's the whole thing.
You multiply the number outside the parentheses by each term inside the parentheses. It works for addition and subtraction inside the brackets.
Most students learn this in middle school and forget it by high school. That's a mistake. The distributive property shows up constantly in algebra, factoring, and solving equations. You need to know it cold.
Why Bother With This?
You might wonder why you can't just compute 4(7 + 3) as 4 × 10 = 40. You're right, you can. But what about 4(7x + 3)? You can't add 7x and 3 together. The distributive property is the only way to multiply a coefficient by a sum that contains variables.
It also makes mental math easier. Instead of computing 6 × 99 in your head, you can do 6(100 - 1) = 600 - 6 = 594. Cleaner, faster, fewer errors.
The Formula Breakdown
The distributive property formula is:
a(b + c) = ab + ac
Where:
- a is the coefficient outside the parentheses
- b and c are terms inside the parentheses (numbers, variables, or both)
The outside factor distributes to each inside term. That's the core idea you need to memorize.
How to Multiply Using the Distributive Property
Step 1: Identify the Outside Factor
Look for a number or variable directly next to parentheses with no sign between them. That number is your multiplier. In 5(x + 2), the outside factor is 5.
Step 2: Multiply the Outside Factor by Each Inside Term
Take your outside number and multiply it by the first inside term, then do the same for the second term. Keep track of any variables.
Step 3: Simplify
Combine like terms if you have them. If you're working with just numbers, calculate the final result.
Numeric Examples
Example 1: 3(4 + 5)
Multiply 3 by 4, then 3 by 5:
3(4 + 5) = 3(4) + 3(5) = 12 + 15 = 27
You could have done 3 × 9 = 27. Same answer. The distributive property gives you options.
Example 2: 7(10 - 3)
7(10 - 3) = 7(10) - 7(3) = 70 - 21 = 49
Works with subtraction too. Just distribute the negative sign along with the coefficient.
Example 3: 6(20 + 5)
6(20 + 5) = 6(20) + 6(5) = 120 + 30 = 150
Or just 6 × 25 = 150. The property holds either way.
Algebraic Examples
Here's where the distributive property earns its keep. With variables, you can't combine terms inside the parentheses.
Example 1: 4(x + 3)
4(x + 3) = 4(x) + 4(3) = 4x + 12
That's your answer. Keep the variable attached to its coefficient.
Example 2: 2(3x + 7)
2(3x + 7) = 2(3x) + 2(7) = 6x + 14
Example 3: -3(2y - 5)
Watch the negative sign here:
-3(2y - 5) = -3(2y) + (-3)(-5) = -6y + 15
The minus sign becomes part of the distribution. Be careful with double negatives.
Example 4: 5(2x + 3y - 4)
5(2x + 3y - 4) = 5(2x) + 5(3y) - 5(4) = 10x + 15y - 20
Three terms work the same way. Multiply by each one.
Distributive Property vs. Direct Computation
Here's a side-by-side comparison showing when each method makes sense:
| Scenario | Distributive Property | Direct Computation |
|---|---|---|
| Numbers only, easy to combine | Unnecessary extra step | Best choice |
| Variables in parentheses | Required | Not possible |
| Mental math (near numbers) | Useful shortcut | Harder mentally |
| Factoring expressions later | Foundation skill | Not applicable |
Common Mistakes to Avoid
Forgetting to multiply every term: If you have 3(x + y + z) and only multiply 3 by x, you've failed. Multiply by all three terms.
Dropping the variable: 2(3x) = 6x, not 6. The variable travels with its coefficient.
Screwing up negative signs: -2(x + 3) = -2x - 6. The negative distributes, not just the 2.
Adding instead of multiplying: 5(x + 2) is not x + 10. It's 5x + 10. You're multiplying, not adding.
Getting Started: Practice Problems
Try these. Cover the answers, work them out, then check.
1. 2(6 + 4) = ?
Answer: 2(6) + 2(4) = 12 + 8 = 20
2. 5(3x + 2) = ?
Answer: 15x + 10
3. -4(2y - 7) = ?
Answer: -8y + 28
4. 6(2a + 3b + c) = ?
Answer: 12a + 18b + 6c
5. 8(9 + 1) = ? (mental math shortcut)
Answer: 8(10) - 8(1) = 80 - 8 = 72
Where This Leads
The distributive property is not an isolated skill. It connects directly to factoring, which is reversing the process to pull out common factors. It shows up in polynomial multiplication. It simplifies solving equations with parentheses.
Master this now. Every algebra topic that follows assumes you can distribute without thinking about it.