Combining Numbers and Variables When Factoring

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  • 0:02 Factoring
  • 1:05 Combining Numbers & Variables
  • 2:48 Example #1
  • 4:34 Example #2
  • 6:10 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.

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 2x + 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 + 3x, 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 18a^3 + 9a^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.

Combining Numbers and Variables

Okay, 18a^3 + 9a^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 9a^2.

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

Example #1

Let's try one with two variables and three terms: 24pq^2 + 32p^3q + 8p^2q. 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 8pq.

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

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