 This video is called factoring polynomials. You can see here we have a polynomial with three terms, 3x to the third, negative 12x squared, and a positive 15x. And we have to just go one step further from the previous video to factor it. Think of it as dividing things out because we want to make things simpler or smaller. It would be another way to think about it. There's simply two steps. Step one, you'll find the GCF, which is what you did in the previous video. And step two, we just have to figure out what's left after we take out that greatest common factor. So for the GCF, remember x to the third is xx and x. x squared, you have two of them and 15x, you just have one. Then ask yourself what number, what's the biggest number that could divide out evenly from 3, 12, and 15? Well, it would be a 3. And how many pairs of x's do I have to pull out? One from each group. And if I try to do a second from each group, I get stuck right there. So 3x would be the GCF. Step two says to put the GCF in front of the parentheses, so 3x in front of parentheses, and then put inside what would be left. So put the GCF in front of parentheses and put inside the parentheses what would be left. Well, how many terms did we start with? One, two, three. So you need to make sure it's really important that when you are factoring, you have three slots or three placeholders, you could say. So now think about it. Three times what, which we'll put right here, would get me back to three. Well, three times one. And now if I've got an x, that's really like having x to the first power. So how many x's if I've got one x right here? How many x's do I need here to get back to x to the third? You need two of them. So you'd write x squared. Two plus one gets you back to that three. Alright, then ask three times what, which will put me right back here, will get me back to a negative twelve. Well, three times a negative four would give me negative twelve. And how many x's do I need in this case? If I have one here, how many do I need right here to get me back to x squared? You need one of them because the one plus one gets you back to the two. Alright, we're almost done. We just have to do this last placeholder. And simply ask yourself three times what gets us back to a positive fifteen. Well, if you thought positive five, you're on the right track. And then here, I already have an x. Do I even need to put an x here? I don't think I do because I already have one x and I only need to get back to one. So I actually won't put anything there. So this would be my factored answer. Now you can check your work. You can know you've done it right if you'd apply the distributive property and see if you get back to the beginning. Three x times one x squared is three x cubed. Three x times a negative four x is negative twelve x squared. And three x times five is a positive fifteen x. If you get back to where you started, so if this answer is where we started, we know we've done it right. And it looks like these two match so we know that this is our factored answer. Let's try this one. Eight x squared minus twelve x. Well, step one is find the GCF. Well, x squared, there's really two x's, negative twelve x, there's one. What's the biggest number that can divide out of eight and twelve? Well, I think that would be a four. And then how many x pairs can come? One x pair. Now we have to fill in what's left. Remember, since we started with two terms, we need place holders for two terms. And now let's figure this out. Four times what, which I'll put right here, would get me back to eight. If you thought two, you're on the right track. And now remember this guy is really x to the first. So how many x's do we need? If we've got an x here, how many x's do I need here to get back to x squared? You need one. Alright, then ask four times what gets me back to a negative twelve? Well, it would be four times a negative three gets me back to negative twelve. Since I already have an x here, I don't need to put another one here because my final answer only needs one and I already have one. So this would be the factored form of eight x squared minus twelve x. I want to be thorough and check my work, so I'm going to do the distributive property. Four x times two x is eight x squared. Four x times a negative three is minus twelve x. Since this matches with where I started, I know I factored it right. It's kind of like you start with your problem. You factor or divide everything out so you broke it apart and then you put it back together again and by multiplication it's a circular thing. It's a way to check to see if you've done this right. Alright, this one, six m cubed. It'll be m, m, n, m minus twelve m squared. That'll be two of them minus twenty-four m which is just one. So I can see I only have one set of m's to pull out. Then ask yourself what number can divide out of a six, a negative twelve, and a negative twenty-four easily? What's the biggest number that could do it? It would be a six. So the GCF is six m. Now I have to fill in my parenthesis with what's left. Since I started with three terms, I need place holders for three terms. So hopefully you're starting to see the pattern. Six times what gets me back to six? It'd be six times one. If I have one m here, because remember this is m to the first, how many m's do I need to get back to m to the third? Be m squared. Six times what gets me back to negative twelve would be a negative two. And how many m's do I need? I've got one here. So how many do I need here to get back to m squared? I would just need one. And then to finish, six times what gets me back to a negative twenty-four would be minus four. And since I already have one m here, and I only need one, we leave this blank. So this would be my factored answer. And I'm going to go ahead and check it. So I'm going to do the distributive property. Six m times one m squared would be six m cubed. Six m times a negative two m will be a negative twelve m squared. And six m times a negative four is negative twenty-four m. That gets me back to where I started so I know I factored everything correctly.