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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.

With an expression, you know how to factor out a number, and you know how to factor out a variable. In this lesson, we'll learn how to factor out combined numbers and variables.

Did you ever mix paint colors, like yellow and blue to make green? Once you add blue to yellow, it's pretty much impossible to ever get it back to yellow.

In algebra, though, that's not the case. We can factor an expression. **Factoring** is the process of finding the factors. Let's say we have 2*x* + 4. We can factor out a 2 to get 2(*x* + 2).

It's like we magically extracted the blue from the green to make yellow. Likewise, if we have *x*^2 + 3*x*, we can take out an *x* to get *x*(*x* + 3). That's like taking the yellow out to make blue.

But what if we have something like 18*a*^3 + 9*a*^2? This one has both numbers and variables that can be factored. Whoa. Our artist's palette just got a bit more complex. We went from finger paints to, well, an actual palette. In this lesson, we'll learn how to combine numbers and variables when we factor.

Okay, 18*a*^3 + 9*a*^2. Let's do this.

If we just wanted to factor numbers, we'd look at that 18 and 9 and find the greatest common factor. That's the largest number that's a factor of both numbers. With 18 and 9, it's 9.

If we just wanted to factor variables, we'd look at that *a*^3 and *a*^2. We can factor out an *a*^2. Why? Because when we multiply terms with exponents together, we add the exponents.

So in this expression, we can factor out both a number and a variable. That means we want to combine the things we're factoring out. That's the 9 and the *a*^2. We put them together to get 9*a*^2.

So, 9*a*^2 is the greatest common factor of the terms in this expression. If we pull out 9*a*^2 from 18*a*^3, what are we left with? Well, the 18 becomes a 2, since 9 * 2 is 18, and the *a*^3 becomes just *a*, since *a*^2 * *a* is *a*^3.

So, we have 2*a*. With 9*a*^2, it all goes away, doesn't it? Well, not quite. We're left with a 1. Why? Let's check our work. You wouldn't paint a picture and not look at it, right? When factoring, it's always best to look at the finished product.

If we say the factored expression is 9*a*^2(2*a*), what happens when we distribute that 9*a*^2? We just get 18*a*^3. Where did our second term go? That's why we need to have not just 2*a*, but 2*a* + 1. So, our factored expression is 9*a*^2(2*a* + 1). It's like we pulled out the mauve from the burnt sienna.

Let's try one with two variables and three terms: 24*pq*^2 + 32*p*^3*q* + 8*p*^2*q*. We graduated to Impressionism here. Let's take this masterpiece apart.

Again, start with the numbers. 24, 32 and 8. Let's see. They're all even, so they all have a 2 as a common factor. They're all also multiples of 4. But, can we get bigger? Yes. 8 is a factor of each number.

What about the variables? We can take these separately. There's a *p*, a *p*^3 and a *p*^2. So, we can factor out a *p*. With the *q's*, there's a *q*^2 and then two *q's*. So, we'll factor out a *q*. That's an 8, a *p* and a *q*. Put it together, and we have 8*pq*.

If we factor 8*pq* from 24*pq*^2, we have 3*q*. If we factor 8*pq* from 32*p*^3*q*, we have 4*p*^2. Let's check that.

8*pq* * 4*p*^2. 8 * 4 is 32. *p* * *p*^2 is *p*^3. And, well, *q*. So we get 32*p*^3*q*, which is what we started with. That's great! Finally, if we factor 8*pq* from 8*p*^2*q*, we're left with just *p*.

So, our factored expression is 8*pq*(3*q* + 4*p*^2 + *p*). I like to think of factoring like watching a painter in reverse. Sure, you take apart the masterpiece, but everything's much cleaner in the end.

How about one more? -8*a*^2*b* - 4*ab*^2 - 16*ab*. With all those negative signs, this is like Picasso's Blue Period - sad, but still pretty great.

Let's start with the numbers. We have 8, 4 and 16. Our greatest common factor is 4.

But wait. That's not all. We have -8, -4 and -16. They're all negative! So we can factor out that negative sign, too. That means that our greatest common factor for the numbers isn't 4, but -4.

It's like we're taking the sad out of the artist. Maybe we got him or her some ice cream. That always makes me happy.

Okay, what about the variables? They all have an *a*, but only one term has an *a*^2. And they all have a *b*, but only one term has a *b*^2. So, we can factor out an *ab*. Put that together, and we're factoring out a -4*ab*.

With the first term, we're left with positive 2*a*. -4*ab* * positive 2*a* is -8*a*^2*b*. With the second term, we're left with positive *b*. -4*ab* * positive *b* is -4*ab*^2. And, finally, the third term is just going to be positive 4. -4*ab* * positive 4 is -16*ab*.

So, our factored expression is -4*ab*(2*a* + *b* + 4). And that's it! Our expression is blue no more.

To summarize, when we factor an expression, we're just finding the greatest common factors in each term. If we can factor both numbers and variables, we look for the factors of each, then combine them into a single term.

To check your work, just distribute the term you factored out. You should get back to your original expression. If all the terms are negative, you can also factor out that negative sign, which makes your expression much more positive.

The knowledge obtained from this lesson could help you to:

- Understand the factoring process
- Recognize greatest common factors
- Identify the numerical and variable factors of an equation
- Factor an equation and combine the factors into a single term

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

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