 Alrighty, welcome back. In this video, we're going to talk about the sum and the difference of two cubes. The sum and the difference of two cubes. Not what I've done, as I've done the liberty of actually writing out the sum of the difference of two cubes, actually the formulas that we are going to use for the sum of two cubes, the difference of two cubes. Summs and differences of cubes have a special pattern that can be used to more easily factor. So there are some type, there are some polynomials that are set up in such a way that they are actually, there's a pattern to how we are going to factor them. And then we call them the sum of two cubes or the difference of two cubes. Now actually, these examples that I have here, now that I'm thinking of it, I'm actually going to switch these around. So I'm going to move this one up here, move this one here. Leave it like that. So now, when we're doing these type of examples, when we're factoring these types of examples, we need to have, we need to consciously think, OK, when I'm looking at these, is this a sum of two cubes or difference of two cubes? And is that going to help me when I factor this problem? So, OK, so now, when I have a sum of two cubes, what that means is I have a cube or I have a number that's being cubed that's being taken to the third power plus another number that's being taken to the third power. Or a difference of two cubes is kind of the same thing, except for difference means subtraction. So a number being cubed, subtract a number that's being cubed. So to look at these examples down here, this right here, 125 d cubed, both of these numbers are cubed numbers. This 125, that's five to the third power. d to the third power is obviously cubed minus eight. I could actually rewrite this as two to the third power. Two cubed is eight. So this actually here is a difference of a cube number and a cube number. So this is actually a difference of two cubes. Now, the pattern that's associated with this is as followed. If I have a sum of two cubes, then that's always going to be equal. The factoring is always going to be equal to those two numbers in parentheses. My first factor is simply just going to be a plus b. It's simply just going to be the number that's being cubed plus this other number that's being cubed, not the end results. And I'll show you a couple examples here in a minute. Not them actually being cubed, but the smaller number. Add those together. And then in the second parentheses is going to be a squared minus a b plus b squared. So it's going to be these individual numbers. We're going to take one of them in the square. We're going to take the a number in square minus a times b here in the middle, and then plus b squared, that number there at the end. And then it's the exact same thing with the difference of two cubes, except now there's a reason why I put these on top of one another is because notice with the difference of two cubes, we have a subtraction here first. And then we're adding everything in here in the second parentheses. Whereas with the sum of two cubes, this first one is adding. But then once we get in here, this first little bit right here, we're actually subtracting this a b part. That's really the only big difference. If you can remember, if you can remember one of these, you can remember both of them because these basically just switch around. Or you can actually think of it this way. This sign here and this sign here switch around depending if it's a sum of two cubes or if it's a difference of two cubes. You can see that switching around. Anyway, so let's actually do an example here to see how we could use these. So right here, this, I could rewrite this as a sum, or excuse me, not a sum, as a difference, because I'm doing a subtraction here, difference of two cubes. As I said earlier, this is 5 to the third power and d to the third power. I could rewrite this as 5d to the third power. If I cube this, I would get back to what I have here. Minus, and I'm going to do the same thing to the other side, 2 to the third power. Now what this does is this gives me what my a's and what my b's are. The a and the b is not 125 and 8. That's incorrect. It's one of a lot of mistakes that happen. In this case, my a number is 5d, and my b number is 2. In this case, so those are actually the smaller numbers that I'm going to use. I'm going to take these numbers, and I'm going to plug them into this little formula, and this is going to help me to factor. Now here we go, I'm going to factor for the first time for this. Now my first parentheses is simply just going to be a minus b. So in this case, it's going to be 5d minus 2. 5d minus 2. There we go. And in the second parentheses is going to be, in this case, a squared, which is 5, I'm going to put 5d squared, plus a, which is 5d. Let me get rid of this parentheses and a little preemptive parentheses there. My a there is 5d, and my b is just 2. Plus, in this case, b squared, my b number is 2. And that's the entirety of my second parentheses. Now I listed out all of that, all the different little plug-in, the numbers, that kind of stuff, so you can see where everything goes. Notice that it is not 125, it's not 8. We're not plugging any of those numbers in. All the numbers that I plug in are 5d or 2. Those are the numbers that I use. So I'm going to simplify this just a little bit. In this case, this first part is just kind of has is. There's no simplifying to be done. This second part, though, this is going to be 25d squared. Square to the 5, square to the d. Plus 5d times 2 is a 10d, plus 2 squared is 4. So that is the second part of that. So that right there, that is an example of a difference of two cubes. How to factor a difference of two cubes. And that's the first example. Now for the second example, it's a little bit more complicated. That's why I switched them around at the very beginning. On the second example, what we're going to do is we're going to, this is a sum of two cubes. It's kind of difficult to actually see that, to see that it is a sum of two cube numbers. Because I don't see any cube numbers here. 108, I don't think that's a cube number. Nowhere is 4, plus this is to the fourth power, not to the third power. But notice that both of these numbers have something in common. They have, let me move them up just a little bit so I can get some room. They have a 4 and an x in common. Both these numbers are divisible by 4. And both these numbers are divisible by x. So what I'm going to do first is I'm going to factor out a 4x. It's what we call monomial factoring. I'm going to pull out, I'm going to factor out a single term first. Now what this does, this gives me x to the third. Plus, 4 goes into 10, twice, 28, would be 27. OK, 108 divided by 4 is 27. OK, now notice though, notice that I have x cubed plus 27. Now 27, we can rewrite this as, we can rewrite this as 3 cubed. So what I'm going to do is I'm going to write that down for x, x to the third plus 3 cubed. Now it's quite obvious to see that this is a sum of a cube number and a cube number. So this is the sum of two cubes. So what I'm going to be doing is I'm going to be using this factoring right here. Now this is a little bit more complicated because I do have this 4x. But honestly, that guy just hangs out at the front. That's all he does. He just sits out front. He's just going to sit there while we do the factoring for this inside parentheses. So that's what I'm going to do. I'm going to start with the factoring. So this 4x, that guy just stays right in front. Doesn't really do a whole heck of a lot. Now the first part here, my first parentheses is simply just going to be a plus b. So now I'm going to identify the a's and b's. In this case, my x is just the a. My b number is just going to be that 3. So now it's just going to be x plus 3 inside my first parentheses, just a plus b. Now my second parentheses, this word gets a little bit more complicated because now I have to take the a number and square it. In this case, my a number is x. So I just take x and square it. My second number here is minus a times b. So I'm going to try to do this all in one step. I'm going to take a times b, which is just 3 times x. 3 times x or x times 3, same difference. So in that case, there's my minus a b part right there. And then last but not least, I have my plus b squared. So over here, my b number is 3. So this is going to be 3 squared, which is 9. And there we go. And that is basically it. That is all the factoring that we can do for those ones. Notice again, I know it's a little bit more complicated because we don't see right away that this is going to be a sum of two q's. But we just take out this 4x. We see there's a sum of two q's. We break it down a little bit so we can see the a's and the b's. And then we actually factor it. This 4x, you just leave this guy out front. And then we got the two parentheses right here. We got the a plus b. And then we got the a squared minus a times b plus the b squared right there. All right. Those are the two factor. And that's how you factor using the sum and the difference of two q's. All righty. Hope you enjoyed the video, and we'll see you next time.