 This video is talking about special products. So when we multiply by two binomials, there's a special thing called FOIL. You might have heard of it before. And that's just a way to remember what you've multiplied together. So it says take the first terms. So we're going to take this x plus 5, x plus 4, and multiply our first terms. And you can see here that's our x and that gives us x squared, which is right here. Then it says do the outside terms. That's the x and the 4. So I'm going to go under to get over to my 4. One was over, one was under. So I have x times 4, which is 4x, which is what we have here. And then we're going to take and do the inside terms. So underneath here, I'm going to put my 5 times my x, and that's what we have here. And then the last terms would be the 5 times 4. And 5 times 4 is going to be 20, which we have here. Now everything's been multiplied. You can see that the x got multiplied by both things with the black. And the 5 got multiplied by everything in the other polynomial with the red. And we end up when we combine our like terms here with x squared plus 9x plus 20. There's another method. Remember we multiplied using a box where we can multiply doing a box with binomials as well. And one polynomial goes up and down, one polynomial goes side to side. We're doing the same problem that we just did. So we should know what our answer is going to be. So remember you take and multiply this side box times the top box. That's x squared. And then we take the side box of x and multiply that times the 4 and we get plus 4x. And we take go down to the bottom row and we take the side of 5 times the x and that gives us plus 5x. Don't forget to take the sign in the box. And then we have the side box of 5 and the top box of 4. 5 times 4 is a positive 20. And again, since they were the same kind of polynomials, x and constant, then my like terms are on a diagonal. And I just have the answer of x squared. And then my diagonal gives me plus 9x plus the 20, just like we had in the slide before. So let's practice. Let's do FOIL first. So the first terms are going to be x times x. So that gives me x squared. The outside terms are going to be x times 3. And x times 3 would be 3x. And then the inside terms, which is the i, that gives me these two terms, negative 8 times x would be minus 8x. And finally, the last terms, the only ones we haven't multiplied yet, negative 8 times 3 would be minus 24. And combining my like terms, then I'm going to have x squared. 3x minus 8x would be minus 5x. And then minus my 24. Now we have a couple of special products that we need to talk about. First of all, is the square of a binomial. It'll look something like a plus b quantity squared. It is two binomials because of the square, but it only shows you one. And a is always going to be the coefficient on x, and b is always going to be some integer. So one thing you need to know, and this is so important, because I have students in all types of classes that multiply a binomial squared and don't get a trinomial. That means three terms, remember. So we need to make sure that we're getting three terms when we get our final answers here. And there is a distinct pattern. If we multiply the first terms, let's come down here and just look at it. a plus b times a plus b. If we multiply the first terms, like Foil tells us to, we get a squared, which is what this says. If I multiply the outside terms and the inside terms, they both give me a b. So how many a b's do I have? I have two of them. So this simplifies to two a b, which is what this says. And then my last terms are always going to be that last term times itself, since I'm multiplying the polynomial by itself. So it's going to be the last term squared. So if it's a plus b, I'll end up with a squared plus two a b plus b squared. And if it's a minus in the middle, well, the a's will always be positive. And even if it's a negative times a negative, I'm going to end up with a positive last term. So it's just the middle term that gets that two and a negative in there, negative two a b. So let's try. Here I have x plus two quantity squared. And I know that my a is going to be x and that my b is going to be positive two. So the pattern said square the first term. So my first term is x and I'm going to square it. So so far I know that I have x squared. And then it says take two times your a and times your b. So I have two times my x, or my a, which is x, times my b, which is two, and two times two would be four times x. So I know I have plus four x. And then take my b term, the last one, and that's going to be two. It says square that one. So two squared will be four. So x squared, x plus two quantity squared is really x squared plus four x plus four. Let's try one more time. So I have x and that's going to be my a. And b is going to be my constant here, but take the sign with it. So it's negative four. And this is x. Make sure it looks like an x. So, square the first term. First term squared was an x is my first term and I square that I get x squared. Then it's two times the a, which is x, times the b, which is negative four. And I get minus eight x. And then it's negative four and I have to square that. And negative four squared will be plus sixteen. Now, I didn't want to caution you that if this had been a two x squared, I would have had to square the two. Okay, let's just try one real quick. Two x squared plus one quantity squared. So I take my two x squared and I have to square that. This is, I'm just going to do the first term because that's the hard one. So two x squared, remember you square everything. So two squared would be four x squared squared. We multiply those exponents and it's going to be four x to the fourth. Okay, so be careful when you have a number in there. Finally, we want to talk about the product of conjugates. And you may be wondering what in the world are conjugates? Conjugates are binomials with the same terms connected by opposite signs. So I have a plus b and I have a minus b. Same terms, the only difference is the sign in the middle. And when I multiply those, I end up with a squared minus b squared. Here's my a squared. And if I do the outside terms, that's going to give me minus ab. And if I do my inside terms, that's going to be plus ab. So what do you notice? They cancel each other out. And that's why we just have the a squared minus the b squared. Positive b times negative b is going to be negative b squared. So find the square of the first term, then find the square of the last term in the conjugate, and then make a subtraction out of it. And you can verify with FOIL if you're not sure that you've done it correctly. Okay, so the first terms here are a. And my second terms are nines. But if I square the first one, I'm going to get a squared. And if I square the nine, I'm going to have 81, nine times nine. And then it says, step three, make a subtraction out of those two. And if I FOIL it, I'm going to have a squared. And then plus nine a. Inside will be minus nine a. And then negative nine times positive nine is negative 81. And my a terms cancel out because they're opposites. There's my a squared minus 81. My first terms are x. And my second terms are 7y. So if I square this first one, it's going to be x squared. And if I square the second one, remember it's 7y that I'm squaring. So that'll be 7 squared is 49. And y squared is y squared. And then we make a subtraction out of it. And we have our solution.