Welcome back, everyone. So earlier in videos, we saw how to multiply polynomials. And when multiplication problems fit a special pattern, like, let's say, I had x+3 and x-3, I could use a special product formula and basically get to the right side. And so this was like x-x2-9, which is a difference of squares.
In this video, we're basically going to do the opposite. So what we're going to see in problems is we're going to see stuff like x2-36, and we have to tell whether it's going to be one of these special products here on the right side. And if we can match it to one of these formulas on the right side, then we can factor and we know that these are just going to be our factors over here. So basically, we're just using these special product formulas in reverse. Alright? So we can also use these things to factor. That's really all there is to it. I'm going to show you how to do this. Let's go ahead and take a look at some problems.
So we have x2-36. So in this case, what happens is my problem has 2 terms. And if you remember from your special products formulas, what happens is we have a couple of things that end up being 2 terms on the right side. There's also a couple of things that end up being 3 terms. So all you have to do here is if this is one of our special product formulas, we're going to look to see whether it's one of the ones that have two terms on the right side. So we're basically just trying to match this thing to one of these formulas. Alright? So does this actually fit one of our special products? Well, I've got x2-36. I have a perfect square, which is x2, and then I have 36, which is another perfect square. These formulas over here deal with perfect cubes, like a3 and b3, so it's not going to be one of these. So it turns out this whole thing is actually a2-b2. That's the pattern. So that means that this factor is down to a+b, a-b. Remember what we saw from these formulas is that these signs over here were opposite signs. So all I have to do is just figure out what my a and b are and I'm done. So what is a? Well, if x2 is a2, that means a is x. And if 36 is b2, that means that b is 6. It's whatever I have to square to get to 36. So that means that this formula over here is just a+b, a-b. So x+6, x-6, and that's it. I'm done. And if you're ever unsure of whether you've done this correctly, you can always just multiply this expression out and you should get back to your original x2-36. Alright? And if you do that, you'll see that that actually does happen. So that's all there is to it. Let's take a look at the second problem here.
Here, we have x3-27. Same idea. I've got 2 terms over here. Let's try to match it to one of these equations here that have two terms on the right side. So x3 is not a perfect square, but it's a perfect cube, and 27 is not a perfect square, but it is a perfect cube. So we're not going to use the difference of squares formula. We're actually going to use these new equations here that we haven't yet talked about, but it's basically, you know, very similar when we have differences of cubes. So this actually turns out to be a3-b3. And so what happens is I'm actually just going to show you what these formulas turn out to be. When you factor them, you're going to get 2 factors of binomial and trinomial. And I'm just going to show you how the signs work out. This is going to be plus, minus, and plus, and this is going to be minus, plus, and plus. You'll almost never have to remember these, but just in case you ever do, one way to remember it is that the last sign should always be positive. These two signs should always be opposites of each other, and the first sign should always be the sign of your original expression. So here what happens is I have an a-b squared, So my first term is going to be a-b. Same sign. And then this other term here is going to be a2+ab+b2. Okay? Now all I have to do here is just figure out what be my a and b are. Well, if x3 is equal to a3, that means that a is x. And if 27 is b3, then b is just a number that when I multiply it by itself 3 times gives me 27. And it turns out that's 3. 3 times 3 is 9, 9 times 3 is 27. So what does this work out to? Well, this just becomes a-b, some other words, x-3. And then the second term becomes a2, so that's just x2. And then I have plus ab, so in other words, these two things multiply together, which is 3x. And then b2 is just 9. So that's it. That's how this factorization happens. You would never be able to do this if you try to do this by, you know, greatest common factor or grouping or stuff like that. So these special products are really helpful. And again, if you want to multiply this out just because you're unsure, you're going to see that a lot of terms will cancel, but you will end up with x3-27 when you're done. Okay?
Last one over here, we have x2+12x+36. So here, what I have here is I have a term or a polynomial with 3 terms and not 2. Remember, we use these equations here when we had 2 terms, or we try to match it to one of these that had 2 terms, but now I actually have 3 terms. So if you remember, we actually talked about some other special products called perfect square trinomials. That was these kinds of equations over here where you had a2+3b+b2, stuff like that.
So which one of these is it? Well, it looks like I have 2 plus signs over here, so I'm going to try to match it to one of these. The one this one that has 2 plus signs. So let's see if it actually works out. Does this actually fit a2+2ab+b2? If it is, then it turns out that we can actually just factor using this formula. If it's not, we're going to have to use something else. Okay? So does this fit? Well, if x2 is a2, then I'm going to guess that a is my x. And if 36 is b2, then I'm going to guess that that b is my 6. The tricky part in these kinds of in looking at these types of trinomials is figuring out if this middle term actually does fit the 2 ab pattern. So let's find out. So this is x2, and then is this 2 times x times 6? And in fact, what actually happens is 2 times x times 6 does, in fact, get you to add to 12x. So, basically, all it's saying is that I've identified this as being a perfect square trinomial. This is really important because if you had had something like x2+10x+36, now this actually doesn't fit the pattern. These are perfect squares, but this one isn't, or this one doesn't fit the pattern. So you wouldn't be able to use this formula for something like that.
Okay? Really importantly, you try to match that. Okay. So what does this become? This just becomes a+2. So what's my a