Introduction to factoring higher degree monomials | Algebra 2 | Khan Academy

In this video, we're going to dig a little bit deeper into our knowledge or our understanding of factoring. Now, factoring is something that we've been doing for many years now. You can go all the way back to when you're thinking about how would I factor the number 12. Well, I could write the number 12 as 3 times 4. I could also write it as 2 times 6. These are all legitimate factors. Or I could try to do a prime factorization of 12, where I'm trying to write it as the product of—you could view it as its most basic constituents—which would be the prime numbers.

So we've done stuff like, well, 12 can be expressed as 2 times 6. 2 is prime, but then 6 can be expressed as 2 times 3. And so 12 could be expressed as 2 times 2 times 3, which we see right over here. This is all review, and this would be a prime factorization. We saw an analog when we first learned it in Algebra One.

In Algebra One, we learn things like—and sometimes this might be in a Math 1 class or even in a pre-algebra class—you'll learn things like, hey, how do I factor x squared plus 6x? And you might recognize that, hey, x squared could be rewritten as x times x, and 6x, that really just means 6 times x. And so both of them have x as a factor. And so we might want to factor that out, and so we could rewrite this entire expression as x times (x + 6).

What we just did is we factored out these x's that I am circling in blue. So in general, this idea of factoring, if you're thinking about numbers, you're writing one number as the product of other numbers. If you're thinking about expressions, you're writing an expression as the product of other expressions.

Well, now as we go a little bit more advanced into algebra, we're going to start thinking about doing this with higher-order expressions. So we've done it with just an x or just an x squared, but now we're going to start thinking about what happens if we have something to the third power, fourth power, sixth power, tenth power, hundredth power. But it's really the same ideas.

We could start with monomials, which is a fancy word for just a single term. So let's say I had something like 6x to the seventh. What are the different ways that I could factor this? Pause this video and think about it. Can I express this as the product of two other things?

Well, I could rewrite this as being equal to 2x to the third times what? Well, let's see what I have to multiply 2 by to get to 6. I have to multiply it by 3. And what do I have to multiply x to the third by to get to x to the seventh? I could multiply it by 3x to the fourth. Notice 2 times 3 is 6, x to the third times x to the fourth is x to the seventh. We add exponents when we're multiplying, when we're multiplying things with the same base.

But this isn't the only way to factor, just as we saw that 3 times 4 wasn't the only way to factor 12. You could also express this as maybe being equal to x to the sixth times what? Well, we would still have to multiply by 6 then, and then we'd have to multiply by another x. So we could write this as x to the sixth times 6x.

There's oftentimes multiple ways to factor a higher degree monomial like this. There is also an analog to doing something like a prime factorization when you're trying to really decompose or rewrite this expression as a product of its simplest parts. How would you do that for 6x to the seventh?

Well, you could rewrite that— you could say six x to the seventh. Well, that's equal to—we could think about the six first. We know that the prime factorization of six is 2 times 3. Two times three, and then x to the seventh is just seven x's multiplied by each other—so times x, times x, times x, times x, times x. How many is that? That's five, six, and seven.

And so some of what we were doing when I said 2x to the third, what we really thought about is, okay, I had a 2, and then I had x times x times x. And then what do I have to multiply that? Well, I have to multiply that by 3x to the fourth.

As we will see, being able to think about monomials in this way will be useful for factoring higher degree things that aren't monomials—things that are binomials, trinomials, or polynomials in general. And we'll do that in future videos.