 One of the important skills we'll want to be able to use is the multiplication of polynomials. This is a fairly involved subject, so we'll split it up into several parts. The multiplication of polynomials relies on the distributive property once again, and to that end we'll introduce two important verbs. We can use distributive property in either direction because it's an equality. If we see the left-hand side, we can replace it with the right-hand side, or if we see the right-hand side, we can replace it with the left-hand side. Now, human beings being what they are, we use different words depending on the direction that we're going. If we go from the left-hand side A times B plus C to the right-hand side AB plus AC, we're expanding. And here it's useful to keep in mind that the type of expression is determined by the last operation performed. When we go this way, we're going from a product into a sum. And since the thing to remember is that when you're expanding, what you should end up with is a sum. A little bit later on, we'll look at what happens when we try to go the other direction from a sum into a product. But for now, it's worth just remembering that this process is known as factoring. So let's try that out. We have 8x times quantity x plus 7, and the directions say to expand. And again, what that means is we'll want to go from a product into a sum. So our distributive property says that if I have something times something plus something, then I can multiply each of our summands by the outside factor. So each of these summands, x and 7, is going to be multiplied by 8x. Now I can do a little bit of work with these expressions. 8x times x, because there's two factors of x, I can write this as 8x squared. This 8x times 7, because multiplication is both commutative and associative, I can rewrite it in any order that I want. And I want to multiply the two numbers together. So this becomes 8 times 7 times x, and I can do that multiplication. And I get my final answer, 8x squared plus 56x. Well, that was fun. Let's try it again. How about the additive inverse of 3x times x minus 5? So it's again convenient to treat everything as an addition, so that we'll first of all rewrite this x minus 5 as x plus the additive inverse of 5. And now I can apply the distributive property. That's additive inverse 3x times x plus additive inverse 3x times additive inverse 5. That first term becomes additive inverse 3x squared. Multiplication is associative and commutative, so I can bring the two numbers together. Additive inverse 3 times additive inverse 5 gives us 15x. And we can move up to a horrifying expression fairly easily. Again, the first thing that's very useful to do is to rewrite all subtractions as addition of the additive inverse. Our distributive property allows us to multiply each of these terms by this outside factor 3x squared. Associativity and commutativity of multiplication allows us to reorder the multiplications any way that we want. And we want to put the numbers multiplied together. And when we do that, we get our final answer. We could actually take one further step. Remember, we don't like to write plus an additive inverse. There's nothing wrong with it, and it's actually very useful to do so. But it's kind of like wearing white shoes after Labor Day. It's something that we'd prefer not to do. So we can rewrite this plus additive inverse 21x squared y and get our final answer. Now, we can also include an expansion as part of an addition. So let's consider this product 2x times x plus 5 minus 3 times x plus 5. So again, it is inconvenient to deal with a subtraction. So the first thing we're really going to want to do is to rewrite this as plus an additive inverse. And now I have two products I can use the distributive property on. I can multiply the terms out. And I do have these like terms here, this 10x plus additive inverse of 3x, so I can combine those like terms. And that gives me my final answer. And what about something like this? Well, the key is to take things one step at a time. And so what we'll do is we'll take the coward's approach. What does that mean? Well, we take a look at this and say, this is scary. I don't want to see this. And so we'll cover up part of the scary part. Now, the coward's approach only works if covering up the scary part does give us some insight into how to proceed. And so what's worth noticing here is that when we covered up the scary part, what we got looks a lot like the setup for the distributive property, something times a sum. If we can be brave now, something times a sum, I know how to handle that. That's something times 2x plus something times 5. But now we can apply the kindergarten rule, put things back where you found them. In this case, we had an x plus 3 here that we kind of hid. We have to put that x plus 3 back where it belongs inside these sets of parentheses. And in fact, because that set of parentheses showed up here we also need to put an x plus 3 in those places as well. But hey, that's not too bad. I'll use the commutative property of multiplication and rearrange the order of multiplication here. And again, what we have is something we can apply the distributive property to. So we'll apply the distributive property, add our like terms together, and we get our final answer.