Everyone. So we've already seen one method of factoring polynomials, which is by factoring out the greatest common factor. For example, if I had x³ minus 2x², once I wrote out the factor tree, I noticed that x² was common in both of the terms. So, I could take that and pull it to the outside, and basically, that's my factorization. I'm going to show you in this video that sometimes for some polynomials, that's not going to work. You're not going to find one common factor that works for all of the terms. In this situation or this polynomial over here, I can't pull out an x². I can't pull out a number like 2 or 4 because whatever I try to do, it won't work for all the terms. I'm going to show you in this video that we're going to need a new method called grouping to solve these types of problems. They're very similar to the greatest common factor, but there are a couple of differences. Let me just show you how it works here. The first thing I want to do is actually talk about when you use these two different methods and, basically, it just comes down to whether you can identify a common factor that works for all the terms, like, for example, the x² over here, or whether you can't. And, usually, what's going to happen in these types of grouping problems is there's going to be 4 of the terms. So if you see a 4 term expression that you can't identify one greatest common factor for, it's usually a good indicator that it's going to be a grouping problem. Also, what you'll notice is that many of these problems, you'll see numbers that are multiples of each other, like 24 and 8 and such.

Alright? So let me just show you how it works. Basically, the whole idea here is I'm going to take this 4 term expression, and I'm going to try breaking this thing up into 2 different groups. That's why we call it grouping. And the hope is that I can factor out the greatest common factor from each one of the groups. So let me just show you how it works. I'm going to show you a step-by-step process. The first thing you want to do is make sure your polynomial is written in standard form just in case it isn't already. So in this situation, I have x³, 2x², 4x, and 8. So this actually already is in standard form, and I don't have to do anything to it. The next thing you want to do is actually do the grouping. And, basically, what you're going to do is you're going to group terms into pairs, and almost always, you're just going to do the first two and the last two terms. It's pretty much 99.9% of the time going to work. Okay. So, basically, what I like to do is I like to drop parentheses around the first two and the last two terms. And now what I'm going to try to do is I'm going to try to factor out a one term greatest common factor from each one of the groups. Basically, what I'm going to try to do is now that I've split these things off into groups, I'm going to turn them into problems where I just try to pull out a greatest common factor out of each one of them independently. So let's try to do this. In fact, I actually already know what the factorization for this expression is. If I try to do this, it's going to be x² times (x - 2) for the first group. So I notice that the x² is common. I draw the little parentheses, and I can pull the x² out to the outside. This just becomes x² x - 2. Alright. So we've already seen that.

What happens with the second group over here? Well, for the second group, what I notice is that the 4 is a multiple of 8. So when I write out the factor tree, this is just 4 times (x - 2). So 4 times 2. So now what's common in this group? The common item in this group is the 4. So I can take the 4 and move it to the outside of the expression, and what I end up with is I end up with a 4 here and then I end up with x - 2. So that was the third step. You're going to factor out a one term greatest common factor from each group. And if you'll notice what's happened here, we've actually ended up getting the same exact thing in both of the sort of expressions. We've gotten the x - 2 term. So if you think about it, now what happens is if you wrap this whole entire thing in one parenthesis, it's actually like the x - 2 is now actually the common thing for both of them. So that leads us to the last step, which is now you're going to factor out this 2 term greatest common factor out of both of the groups. So it's the same thing I did with the x² and such. I take this x - 2, and pull it to the outside over here, and then just write everything that's inside of parentheses that remains. So this just actually just becomes x - 2, and then you have x² + 4. And this actually is the complete factorization of this polynomial. If you go ahead and foil this out, what you'll actually see is that you'll end up getting back to your original expression over here. Alright? Alright. So, again, just to summarize, if you ever see a polynomial and it's 4 terms and you notice that you can't find a common factor that works for all of them, but you notice that some of the numbers are multiples of each other, try splitting up into 2 groups. And what you're going to see a lot of the time is that you'll very coincidentally end up with the same factor that you can pull out of both of the groups. And so this is a very specific type of problem, but it's good to know. Anyway, so that's it for this one, folks. Thanks for watching.