yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

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


3m read
·Nov 11, 2024

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.

More Articles

View All
Pinstriping my Lotus Exige S240 for $7
What’s up? You think Mornington granted today is going to be a fun game? I’m all the ways of red clay and my God, I wore last week judgment. Lots of luck, and I got this off of eBay, like six dollars in China. What it is, it’s the final lightning pink. I …
Going Inside MEGA Rehab | Explorer
Do ter de made a token attempt to increase capacity by building a mega rehab facility on a military base about four hours north of Manila. Our crew is the first ever to be allowed inside to film. It’s a big complex divided into four phases. Each phase can…
Backspin Basketball Flies Off Dam
Recently, some friends of mine went to the Gordon Dam in Tasmania, which is 126.5 meters (or 415 feet) high. Then they dropped a basketball over the edge. You can see that the basketball gets pushed around a bit by the breeze, but it lands basically right…
Chris Hemsworth sends his best mates in search of the secret elixir of Bali | Azza & Zoc Do Earth
[cheering] - Hello. Chris Hemsworth here. I’ve decided to create a new series about unlocking health and wellness secrets around the world. Here’s the catch. [announcer] Chris Hemsworth! [Chris] I’m too busy to travel to all these countries and get the go…
KAMALA REFUSES TO APOLOGIZE!
I think what he did and how he did it did was did not make much sense because he actually didn’t do much of anything. She missed the opportunity yet again. This thing is killing her, this border wall thing. What she should have come out and said—and she …
How One Man's Amazing Christmas Lights Have Spread Joy for 30 Years | Short Film Showcase
[Applause] [Music] [Music] [Music] [Music] My name is Bruce Mertz, and the people around here call me Mr. Christmas. This is my 31st year of putting up the lights, and I’ve been living here since 1977. Every year, I start setting up at the end of August.…