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Factoring Out Variables: Instructions & Examples

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  • 0:01 Factoring
  • 0:53 Numbers
  • 2:34 Variables
  • 4:02 Exponents
  • 6:00 Lesson Summary
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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.

Let's find the factors. In this lesson, we'll learn how to factor out numbers or variables from an expression. We'll even learn how to factor variables with exponents.

Factoring

You probably know that 10 is 5 * 2. We call 5 and 2 the 'factors' of 10. They're what we multiply together to get 10. It's like how a cockapoo is the product of a cocker spaniel and a poodle. You put a cocker spaniel and a poodle together, and you get this adorable product with a silly name.

Once you have a cockapoo, you're kind of stuck with it. But with a number like 10, you can use factoring to get back to its parents. Factoring, then, is just finding the factors.

Makes sense, right? Fishing is finding fish. Birding is finding birds. And factoring is finding the factors. If only cooking were finding someone to cook for you. In this lesson, we're going to learn how to take not just a number, but an entire expression, and factor out numbers or variables.

Numbers

Let's start with numbers. Here's an expression: 6x + 12. It's a nice expression. It's like a schnoodle, the schnauzer/poodle hybrid. But unlike a schnoodle, we can take 6x + 12 and factor out some of its genes. Please don't try this with your dog at home.

To factor a number out of an expression, we need to find the highest common factor. That's the largest factor shared by all the terms. Here, we have a 6 and a 12. What are the factors of 6? Well, 1 * 6 is 6, so 1 and 6 are factors. 2 * 3 is also 6, so 2 and 3 are factors. And that's it. The factors of 6 are 1, 2, 3 and 6. What about 12? 1 and 12, 2 and 6, 3 and 4. Okay, the highest common factor is the biggest number in both of these lists. Here, it's 6. 6 is our highest common factor.

What happens if we factor out a 6 from both terms? This means we divide each term by 6. 6x becomes just x. 12 becomes 2. We write our factored expression as 6(x + 2). We know we did it correctly if we can work backwards, multiplying the 6 by each term, and get back to 6x + 12.

Also, note that we could have factored out another common multiple, like 3. That would get us 3(2x + 4). But we wouldn't have completely factored the expression. You can't mix part of a schnauzer and a poodle. You'd end up with a schnaup or an oodler. And that's just not right.

Variables

Okay, we factored out a number, what about a variable? This works in much the same way. Here's an expression: xy + 7y.

Note that we can't factor out any numbers. But what is shared by both terms? They both have a y, don't they? So, we can just pull out that y. What happens if we do? The xy becomes just x. And the 7y becomes just 7. So we have y(x + 7).

I think this is like if you take a puggle, a pug/beagle combo, and decide you'd rather have a pug and a beagle. I mean, pugs and beagles are both great dogs, why not keep them factored? Okay, a puggle is pretty cute.

Here's another expression: ab + a. Here, we don't have any numbers. But both terms have an a. If we take an a out of ab, we just have b. So is our answer a(b)? No. Because a * b is ab, not ab + a. We can't lose sight of that second a. You may think a dog's tail isn't very important, but try explaining that to a dog.

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