*Jeff Calareso*Show bio

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

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Instructor:
*Jeff Calareso*
Show bio

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

The distributive property, which involves distributing a term across all numbers and variables within the parentheses, provides a useful way to simplify algebraic expressions. Learn what the distribution property is, and solve for it in algebraic expressions in practice problems.
Updated: 09/24/2021

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.

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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.

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

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

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

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

How about one more? -(5*a* - 3*b*). 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* 5*a* is -5*a*. And -1 * -3*b* is positive 3*b*. So our simplified expression is -5*a* + 3*b*.

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.

When this lesson is finished, you should be able to utilize the distributive property when solving algebraic expressions that require multiplication.

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