 This video is going to talk about multiplying polynomials. So a couple things we need to remember, because they will apply when we're multiplying here, is our same base multiplying property, a to the m times a to the n. Remember, same base, so we add the exponent, so that would be m plus n. Add those exponents. And the distributive property, remember, you had to distribute the a to everything inside. So that gave you a, b plus a, c. We're multiplying monomials, exponent property. And that distributive property is really what we're dealing with. Both of them come right into play. So I have x times x squared. Well, I have two factors of x here, and I'm going to multiply by one more factor of x, same base. So I'm going to multiply and have x to the third. Now when I do this one, we're going to multiply our numbers, and then we're going to multiply our b's, which will be the same base and use our exponent property. So my numbers are 2 times 3 gives me my 6, and then my variables are b cubed and b, so it's 3 plus 1, if you want to see the work for what we're doing. And that gives us a final answer of 6b to the fourth. Now when we multiply a monomial by a binomial, that's what we were doing at the very beginning. We just have to make sure that this 5 gets multiplied by everything inside. So it's 5 times x, and 5 times 1 would be plus 5. Numbers aren't so bad. What happens when we have variables? Well, let's do this step by step. What we really have here is 2z times z, and I'm just going to write it just like that. I have 2z times z, and then I'm going to have a negative 3 times 2z. So I have negative 3 times 2z. So what does that give me? 2z times z gives me the 2, and then the 2 factors of z would be z squared. And then again, we do our numbers. So negative 3 times 2 is negative 6, and then I don't have any other z's, so it's just minus 6z. What happens when we get bigger polynomials? Well, we have to distribute each term in the first polynomial to everyone in the next polynomial. Now again, I have a monomial here, but I've got a messier-looking thing going. If I just do the distributive like I'm used to doing, then I'm going to have 4x squared y, and I'm really going to multiply that times x squared y squared. And then I have plus 4x squared y, and I'm going to multiply that by 3x squared. That's going to the second term. And then I have a positive 4x squared y on the outside, so I'm going to write plus 4x squared y. And then in the parentheses, I'll give me my negative 7y squared. So 4 is the only number, so I don't have to multiply that with anything. x squared times x squared. Remember, we're going to add these two exponents, so add them. So that gives us x to the fourth. And then y times y squared, again, we're going to add those two exponents. So we have y cubed. And this next problem, we've got numbers this time. So 4 times 3 will give me a positive 12. They're both positive. And then I've got x times x. And remember, we're going to add those exponents, so x to the fourth again. And then I don't have any y's on the inside, so it's just carry along the y from the outside. And then this last one, again, we have numbers. So positive 4 times negative 7 is negative 28. And I don't have any other x's, so I'm going to carry my x squared on the outside to get around. And then I talk about my y's, and I can add my exponents, and that will give me y cubed. So my final answer is this 4x to the fourth y cubed plus 12, oops, plus 12x to the fourth y minus 28x squared y cubed. That was a lot. All right, so we have this example here. And we can choose which way we want to go. And I'm just going to go ahead and do the distributive property. So I'm going to take my 6a squared and make sure it gets multiplied by both terms inside this polynomial. 6a squared times 3a squared, 6 times 3 is 18, and a squared times a squared gives me four factors of a. And then I'm also going to take that 6a squared and multiply it by my negative 2. 6 times negative 2 would be minus 12, and I carry my a squared along. Next I'm going to come in and look at my 2a, or negative 2a. So negative 2a times my 3a squared will give me negative 6, and I have three factors of a. And then I take my negative 2 and I multiply it negative 2a and multiply it by negative 2, and I have plus 4 and then carry along the a. And finally, I'm going to take this 5 and I'm going to multiply it by the 3a squared. So that's plus 15a squared. And then I take it, the very last thing is to take the 5 times the negative 2, which will give me minus 10. So now it's time to combine like terms. So let's see what we have. We have an a squared term here and an a squared term here. And I've got nothing else in common. That's the only like terms we have. So let's rewrite the problem. 18a to the fourth, negative 12a squared plus 15a squared would be plus 3a squared. So I've taken care of that and I've taken care of my like terms. Negative 6a cubed plus 4a, that's a 4, and minus 10. Now if you were to look at this in the back of the book, they would put it in descending order. 18a to the fourth minus 6a cubed, then plus the 3a squared and plus 4a minus 10. But I would accept either one of those. Alright, last time we did a nice distributive. Let's try this one with a box just to see what happens with a box. So I need three columns for my first polynomial. So there's going to be my three terms right there. One, two, three. And then I need three. This one never has anything. So there's one term and there's my second term and there's my third term. So let's fill them in. It doesn't matter which one goes where, but each box gets a term from a polynomial and one goes side to side, one goes up and down. So this one is first, so I'm going to go side to side with this one just because I want to. X squared and then the second box would be minus 6x and the next one would be plus 5. And then on the outside, going up and down are going to be my purple ones. X squared is the first term, plus 3x is the middle term, and then minus 8. You need to make sure you put the signs in the box so that all your signs work out right. Alright, here we go. X squared times X squared is going to be X to the fourth. X squared times negative 6x will be negative 6 and then we have three factors of X. And I've got my X squared times my 5 and that will give me plus 5x squared. Now I'm going to do my 3x times the X squared. So that gives me plus 3x cubed. There's three factors. And then I'm going to do my 3x times my negative 6x. So negative 6 times 3 is negative 18 and then X squared. And then I'm going to take my 3x times my 5 and that will give me plus 15x. I'm going to take negative 8 and I'm going to multiply by X squared. So I have negative 8x squared. And then I'm going to take my negative 8 times my negative 6x. And that's going to give me a positive 48x. And then finally I'm going to take my negative 8 times my 5 and that will give me a negative 40. And if you look real carefully I did my like terms in light colors. My like terms end up being on a diagonal. So let's combine our like terms then. X to the fourth is the only one that we have that's to the fourth power. On this diagonal is my X cubed. So 3x cubed minus 6x cubed. So it's negative 6 plus 3 would be minus 3 and then X cubed. And here's a little bit more work here. We have negative 8 and negative 18. So let's do it over here. Negative 8 minus 18 plus 5. Well if I do negative 8 minus 18 that's going to be a negative 26. And then plus 5 and negative 26 plus 5 will be negative 21. So we have minus 21 and those are all X squareds. And then we have 48 plus 15. They're both positive. So that's going to be plus 63x. And then we have minus 40. Wow.