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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.

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.

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.

Let's start with numbers. Here's an expression: 6*x* + 12. It's a nice expression. It's like a schnoodle, the schnauzer/poodle hybrid. But unlike a schnoodle, we can take 6*x* + 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. 6*x* 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 6*x* + 12.

Also, note that we could have factored out another common multiple, like 3. That would get us 3(2*x* + 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.

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

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 7*y* 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.

Remember that a variable on its own is the same as 1 times the variable. So if we're dividing 1*a* by *a*, we get 1. That means that our factored expression is *a*(*b* + 1).

What if we have exponents? Here's one: *y*^3 + 9*y*^2. How do we factor this? It looks like one of those mutts that you can't really be sure what breed it is. Or can you? We know both terms have a *y*. What else do we know?

What happens when you multiply terms with exponents? You add the exponents. *x*^3 * *x*^3 is *x*^6. So *y*^2 is just *y* * *y*. And *y*^3 is *y* * *y* * *y*. If we look at it this way, each term has 2 *y*'s in it, or *y*^2. Let's factor out a *y*^2. If you take a *y*^2 from *y*^3, you're left with just one *y*. And if you take a *y*^2 from 9*y*^2, you get just 9. So, our factored expression is *y*^2(*y*+ 9). Case closed on the mystery of the mutt.

Okay, let's graduate to something more complex: *p*^3*q*^2 + *pq*^3. Oh, gosh. Is this like a Siame-huahua, that unholy mix of a Siamese cat and a Chihuahua? No. Awesome as that name is, the Siame-huahua doesn't exist. Or so the government would like you to believe.

But we're going to do the same thing here. Both terms have a *p*. And both terms also have a *q*. More than that, they both have a *q*^2, just like in that last example. So, we can factor out *pq*^2. If we take *pq*^2 out of *p*^3*q*^2, the *q*^2 factors to 1 and the *p*^3 factors to *p*^2. With *pq*^3, the *p* factors to 1 and the *q*^3 factors to just *q*. That makes our factored expression *pq*^2(*p*^2 + *q*).

To summarize, we learned about **factoring**, or finding the factors. In an expression, we're seeking the highest common factor. This is the largest number or variable shared by all the terms.

When we factor out numbers, we can determine all the factors of each number, then find the largest one that is in each set. When we factor out variables, we likewise find the variable or variables that are shared by all the terms.

When we factor expressions containing variables with exponents, remember that exponents are added together when the terms are multiplied. And, again, factoring expressions in algebra is great. Trying to factor your hybrid dog? Not okay.

By the end of this lesson you should be able to factor numbers, variables, and exponents.

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

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