The Distributive Property and

The Distributive Property and Algebraic Expressions

The order of operations is great. But sometimes we need to bend the rules to simplify an equation. Fortunately, the distributive property gives us a scenario in which this is okay. Learn all about it in this lesson.

Distribution

What do you think of when you hear the term 'distribution center'? I think of the mail. When you mail a letter or a package, you might bring it to the post office or put in a mailbox. People all over your town are doing the same thing. All the items from your town get collected and go to a distribution center.

Once they're there, they get sorted and distributed into different trucks depending on where they're going. Then they leave the distribution center and head off to other parts of your town, your state, or even other parts of the world.

The distribution center is the place where everything is organized into logical groups. All the mail for Wyoming goes in one place, and all the mail for Japan goes in another. And this is essentially what the distributive property is all about.

Distributive Property

The distributive property is a handy math rule that says when you are multiplying a term by terms that are being parenthetically added, you can distribute the multiplication across both terms, then sum their products.

That was totally confusing, I know. The distributive property is much easier to show, and it's much simpler than it sounds. Think of it this way: a(b + c) = (ab) + (ac).

Let's prove it with real numbers. If we have 5(3 + 4), the order of operations tells us we start with the parenthesis. So we do 3 + 4 = 7, get 5(7) and then end up with 35. That's all well and good. But the distributive property tells us that in this situation, we can instead do (5 * 3) + (5 * 4), where we distribute the 5 across the parenthesis. Does it work? We get 15 + 20. Is that still 35? Yep. It is.

For most purposes, the distributive property is limited to multiplication. And while I said that the parenthetical terms must involve addition, remember that something like (7 - 2) is really just (7 + (-2)), so this rule still works.

If you're wondering why (5 * 3) + (5 * 4) is in any way easier than 5(7), well, it isn't really. It would help to look at some examples of when this is particularly helpful.

Practice Problems

What if you can't add what's inside the parentheses? Look at this one: 7(3x + 5y). You can't simplify 3x + 5y. But you can distribute the 7 and get 21x + 35y. In fact, that's when you'll most often use this rule - when you have variables.

Here's another one: -5(6 + 2x) Don't forget that negative sign. If we distribute the -5, we get -5 * 6, which is -30, and -5 * 2x, which is -10x. Put that together and our simplified expression is -30 - 10x.

Here's one with a minus sign inside the parenthesis: 4a(6 - 2a). Remember, 6 - 2a is really just 6 + (-2a), so our two terms are 6 and -2a. 4a * 6 is 24a. And 4a * -2a is -8a^2. So our simplified expression is 24a - 8a^2.

Let's try one that's a little more complicated: -2x(x - 8y). Again, pay attention to those negative signs. -2x * xis just -2x^2. Okay, that's not so bad. And -2x * 8y? Wait - remember, it's -8y. Okay, -2x * -8y. You can't add x+ y, but you can multiply them. We get positive 16xy. So our simplified expression is -2x^2 + 16xy.

How about one more? -(5a - 3b). What's that negative sign hanging out in front of the parenthesis? It's really a -1. So we need to distribute the -1 across the terms. -1* 5a is -5a. And -1 * -3b is positive 3b. So our simplified expression is -5a + 3b.

Lesson Summary

In summary, the distributive property can be expressed as a(b + c) = (ab) + (ac). All we're doing is distributing the a across the terms inside the parenthesis. This is especially useful when we're dealing with variables that can't be added. The distributive property gives us the power to simplify our expression.