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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

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Lesson Transcript

Instructor:
*Jeff Calareso*

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

What's the purpose of distribution in algebra? In this lesson, we'll find out why we distribute. We'll also look at examples and sample problems that involve basic distribution.

Tommy loves to chew gum. He always has a pack of gum in his pocket, much to the dismay of his parents, who sometimes discover it in the wash.

At school, Tommy isn't supposed to chew gum in most classes. But his algebra teacher, Mr. Adams, is cool with it. Mr. Adams tells Tommy he can chew gum in class as long as he brings enough for everyone. He needs to *distribute* his gum to his algebra classmates.

What does that mean? That means that Tommy takes that pack of gum in his pocket and hands out a piece to everyone, even Mr. Adams. This distribution of gum is just like the distribution they're learning about in algebra, only with more spearmint freshness, more loud gum snapping, and more than a few pieces of gum ending up decorating the desks and chairs.

In algebra, **distribution** means to spread out terms equally across an expression. We refer to what we're doing as the **distributive property**, which can be defined as *a*(*b* + *c*) = *ab* + *ac*.

Why do we distribute? It's a way of simplifying expressions. This can make them easier to work with. If Tommy shares his gum with his class, more gum chewing can happen. Likewise, if we distribute the *a* across the parentheses, we've made our expression simpler, and when distribution is part of a larger equation, we can do more with it.

Here's an expression: 6(*x* + 2). *x* and 2 are like students in the class, wishing they had some of that gum. Maybe they had salami for lunch. The 6 is like the pack of gum. We want to distribute it to each of the terms inside the parentheses equally.

To do that, we take the 6 and multiply it by each term. So 6 * *x* = 6*x*, and 6 * 2 = 12. The only operator we have is that plus sign, so we end up with 6*x* + 12. This is our simplified expression. See how much happier our *x* and 2 are? They're quite literally 6 times happier; plus, no worries about grossing out the cute girl in English class.

What if we have an expression like this: 3(2*x* + *y*)? This is a class with two variables. I think a variable in a class is that kid who might be a class clown one day and then sit quietly in the back the next day. He's variable; we just don't know. And a class with two variables? That's a recipe for disaster.

But it doesn't change how we distribute. We still take the number outside the parentheses and distribute it, or multiply it, with the terms inside the parentheses. So, 3(2*x* + *y*) becomes 3 * 2*x* and 3 * *y*.

- 3 * 2
*x*= 6*x* - 3 *
*y*is just 3*y*

So, our simplified expression is 6*x* + 3*y*. Notice that we can't simplify this any further. We can't add the *x* and *y*, so this is as far as we go. It's like cutting up a birthday cake that someone brings to share. You can only cut the slices so small before they're not so much pieces of cake as just crumbs, and then everybody gets sad.

Let's try another: 2(5*a* + *b* + 3*c*). Three variables! What do we do here? Ok, let's imagine Tommy's friend, Steve, brings in cupcakes. That'll make Steve quite popular, because he brought enough to share with all the variables. Distribution with three variables follows the same logic. Just be careful to not lose track of your terms; 2(5*a* + *b* + 3*c*) has a lot going on. Let's take it slowly.

- We start with 2 * 5
*a*. That's 10*a*. - Then 2 *
*b*. That's an easy one. It's just 2*b*. - Then 2 times what? 3
*c*. That's 6*c*.

It can be easy to mix up the variables or the coefficients (which are the numbers in front of the variables). Let's see what we have: 10*a* + 2*b* + 6*c*. Can we simplify any further? Nope. So, this is our final answer.

What if we saw one like this: 2(3*x* + 2*y* - 4*z*)? That's pretty similar to the last one. And, fortunately, Steve still has cupcakes to share. But let's see what's different. One of our operators is a minus sign. That's that kid with black clothes, fingernails, hair and all that. He's kind of negative.

We still distribute just the same, but we keep an eye on that different operator. So, we get 6*x* + 4*y* - 8*z*. We're still dong 2 * 4*z* with that last term, but we include the minus sign in front of it.

Ok, so far, so good. We've successfully distributed with all different kinds of expressions. And then we encounter this: *x*(2 + *y*). Hello. What do we have here? How can we distribute a variable? You don't ask the pack of gum if it brought enough Tommy's to share. That's just weird.

Fortunately, this expression follows the same distributive property as all the others. We treat the variable *x* just as we would if we had a constant, or number, outside the parentheses. We just do *x* * 2, which is 2*x*, and *x* * *y*, which is *xy*. That makes our simplified expression 2*x* + *xy*, and no students were harmed in the distribution of this example.

To summarize, we learned about the power of sharing. Gum, cupcakes, numbers or variables - sharing is caring. Specifically, we focused on distribution, or spreading out terms equally across an expression.

We practiced using the distributive property. This is defined as *a*(*b* + *c*) = *ab* + *ac*. This method of distribution works with any type of expression. It's just like sharing gum - spearmint or cinnamon, strawberry or watermelon, everybody gets a piece!

When you get to the end of this lesson, you might know how to:

- Use the distributive property to simplify algebraic expressions
- Distribute a variable

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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

- Why Do We Distribute in Algebra? - Explanation & Examples 5:39
- Distributing Positive and Negative Signs 5:56
- Distributing Algebraic Expressions with Numbers and Variables 7:57
- Changing Negative Exponents to Fractions 6:24
- Working With Fractional Powers 6:38
- Distribution of More Than One Term in Algebra 6:12
- Go to High School Algebra: Algebraic Distribution

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