Welcome back, everyone. So by now, we've seen how to multiply polynomials by either foiling or distributing. What I'm going to show you in this video is that some multiplication problems will actually fit a special pattern, and when this happens, we can use formulas, these things called special product formulas, to multiply because it's going to be way easier and way faster than having to do stuff the long and tedious way by foiling or distributing. I'm going to show you that you can take a look at a problem like this, and very quickly, you'll be able to tell me that the answer is x2-25. I'm going to show you how it works. So, basically, if I take a look at an expression like this, the only way I know how to multiply it right now is by foiling. That's what I showed you. That's the first and outer, and then do the inner and last. So I'm going to multiply all these terms out to show you. This just becomes x2-3x+3x+-9. Alright? And if you simplify this, what you'll see is that the negative 3x and positive 3x cancel, and all you end up with is x2-9. So that was a lot of work just to figure out that 2 of the middle terms will cancel, and the answer is relatively straightforward. It's just 2 terms. Now what actually happens is the reason this happened is because these two terms are the same, but just opposite signs. And whenever that happens, the middle terms are always actually going to cancel. So this is like a little bit of a pattern. This always happens as long as these two numbers are the same, and so we've come up with actually a special formula for this. And it's basically if you can sort of see if you can detect that a multiplication problem fits the sort of pattern of a+b times a-b, the answer will always just be a2-b2. This equation pops up a lot in math. We actually give it a name. It's called the difference of squares, but you don't need to know the name. So, basically, the whole thing is you can take a look at a problem, and if you can figure out that it fits this pattern, then you could just go straight to this rather than having to foil it, and that's going to be your answer. So, let's take a look here. I have x+5 times x-5. So, in other words, I have the same number, just opposite signs, so this is like a+b, a minus b. So the answer is just going to be a2-b2. So, basically, what happened is that, like, my a is my x, and b is my 5, and so, therefore, what happens is when I take a squared, that just becomes x2, so these two things are the same, and when I do b squared, that just becomes 25. Alright? And this is just your answer. Alright? So, basically, the way that these problems work out is it's kind of like pattern matching. Anytime you see a multiplication problem and you see the same numbers but opposite signs, or you see stuff that's squared, usually, it's going to fit one of these formulas, and you could just go straight here instead of having to multiply everything out the long way. Let's take a look at this next problem here, x+62. Alright. This actually fits a special pattern of a+b squared or ab-b squared. These equations are called the squares of binomials, and I'm just gonna show you how these work out. This is basically what the right side equals, and we have a plus and plus as the two signs and a minus and a plus. One easy way to remember this, by the way, is that the first sign will always be the same as the left side, and the second sign will always be positive. So that's one way you can remember this real quick. Alright. So, if I take x+6 squared, which one of the two patterns is this? Well, it's clearly this one over here, so I'm just going to basically use my formula. So this is a, and this is b. So, in other words, my a is x, my b is 6. So what the answer is supposed to be is the answer is supposed to be a2+2ab+b2. All I have to do is now just replace these letters here with x's and sixes and stuff like that. So my a squared just becomes x2. What about this 2ab? Well, this just becomes 2. Then what's a? A is x. And then what's b? B is 6, and then plus b squared. B squared ends up being 62, which is just 36. So, if you simplify this, what you end up getting is x2+12x+36, and that is how to multiply this, again, very quickly without having to foil this. Alright, folks. So, really, that's all there is to it.