 Hello, today we're going to be working on multiplying polynomials with the box method which means we're going to be doing a very visual way to do this. Now don't forget you can always go back to the ways that I've shown you before with multiplying polynomials by just distributing or expanding as another word here. So you could certainly distribute the x through to both sets of parentheses or both pieces in that parentheses and distribute the 2 to both pieces in the parentheses. But like I said I want to show you a more visual way today to do this. And it kind of helps you keep things organized especially if some of these polynomials get bigger. Okay so what you want to do is go ahead and draw yourself a rectangle and mine looks good enough. Because the first set of parentheses is two terms I'm going to go ahead and split it in half this way and because the second set of parentheses is two terms I'm going to split it in half on the other way. And then what you're going to do is you're going to take and write out your two terms so x and then plus two plus two and at the top I'm going to write x and then minus four. So the side here represents my first parentheses the top represents my second parentheses. Order doesn't matter but I tend to do it this way just to be consistent. So now we're going to multiply these so x times x gives me an x squared. x times negative four gives me a negative four x. x times positive two or positive two times x gives me a positive two x. And positive two times negative four gives me a negative eight. Okay so those are basically my four terms that I get when I multiply these together. So I'm going to end up with x squared minus four x plus two x minus eight. Okay then you'll notice here in the middle just like we talked about again the last time we have like terms here and here. So when I combine those together I'll get a final answer of x squared minus two x minus eight and that would be my final answer. And if you've learned the acronym FOIL you can certainly do that too I mean it doesn't really matter. One thing I will have you notice let me highlight this in a different color is our like terms fall here and here. And that's typically going to be the case they're going to be on a diagonal I can't say always but typically I think that's a pretty good thing that you can recognize. Okay the next problem gets a little bit bigger and a little bit uglier. So if you were to try to do this out you'd have to distribute the x to all three of these terms and distribute the negative two to all three of these terms. You're likely to lose stuff I know I always do. So again I'm going to draw out a rectangle I'm going to make this one a little bit longer since I have three terms instead of two. So because my first set of parentheses is two terms I'm going to divide it in half. And then because this one has three terms I'm going to divide this one into three fairly equal pieces. Again I'm going to write out what I'm multiplying so on the side I'm going to write x and then minus two and then along the top I'm going to write x squared minus three x and then plus one. And you don't want to make sure you lose any of your signs I try to put mine in just to make sure I remember although I never usually put them on the first ones I just noticed. Okay let's go through and multiply each piece so x times x squared gives me an x to the third. X times negative three x gives me negative three x squared. X times positive one gives me a positive one x. Negative two times x squared gives me negative two x squared. Negative two times negative three x gives me a positive six x. And negative two times negative one or positive one gives me a negative two. Okay again let's go through and look for like terms. So I've got a like term here and here and then I've got a like term here and here. So when I combine all of my like terms together the x cube doesn't have one so I'm just going to write that by itself. I combine my negative three x squared and my negative two x squared. I get a negative five x squared. I combine my positive one x and positive six x that gives me a positive seven x. And then I've got my negative two here dangling on the end. So that is our final answer. So you notice we didn't lose any pieces. It's easy to see where our like terms are at. And I think this is just a fantastic way to get a visual representation. Alright the next one I went ahead and set up the box ahead of time just to give you guys a different visualization of it. So here I'm going to multiply since that's what the problem says. So I'm going to write my two x minus one. So you notice I've already got my two set up here at the side and the top. Along the top I'm going to write two x and then plus one. So I'm going to multiply these together. Two x times two x gives me a four x squared. Two x times positive one is a positive two x positive. Negative one times two x gives me a negative two x. And negative one times positive one gives me a negative one. Okay again I've got some like terms on those diagonals. So here and here the other two are not though. So my four x squared I can't combine that with anything. So that's just a four x squared. You'll notice a negative two x positive two x those cancel. So I end up with a negative one. And four x squared minus one is my answer. You notice this only has two terms though. Even though we multiplied two by two. Well a lot of terms went up to three. This time we only end up with two terms. This is called a difference of squares. Keep that in mind because we'll need that when we go to factor. But anyway we'll get to that in a minute. Okay last problem on here multiply with the box method. And this says x plus five quantity squared. Now if this were x times five we could distribute that square. But because of this plus sign in the middle here we cannot. So remember when you square something that means you have to multiply it by itself. So that means I've got x plus five times x plus five. Let me get rid of this little arrow here. Okay so that means I've got a two by two. So I'm going to go ahead and set up my rectangle. And split it into both ways. So on one side I'm going to have my x plus five. Along the other side I'm also going to have an x plus five. So obviously it's the same thing because it's squared. So when you multiply x by x you get x squared. Multiply x by positive five you get a positive five x. Multiply positive five by x you get a positive five x. Multiply positive five by positive five you get a positive 25. Okay again I have like terms on the diagonals. That's why these are so nice they set these themselves up really nicely for you. So I've got my x squared. Can't combine that with anything. I had a positive five x and a positive five x. I get a positive ten x plus my last term here is a 25. And that is your final answer. Okay so hopefully you know again use this if you like it. Don't use it if you don't like it. We are going to be using this with factoring now. And so I think it's a good thing for you to practice just to get an idea of what's going on. Now on the second page here I do have some problems. So the first problem says when you use the box method you should see some patterns. Where in the box is the product of the first terms of each polynomial? Okay so here you'll notice first term first term first term first term. So the way I set up my box it's going to be in the upper left corner. And ignore all these little lines here that must have come from above. Okay so that's the upper left. Where in the box is the product of the last terms of each polynomial? Okay so I've got, let's see, last term last term so that went down here. Last term last term that went down here. Last term last term that went down here. So I think I'd call that the bottom right. So bottom right. Where are the like terms located? Well I think I emphasized that enough. Those are located on the diagonals. And the last one here says what do you notice about the products of the terms on each of the diagonals? So the key word to this one is products. So that means if we multiply our diagonals what do we get? So if I multiply an x squared by a 25 I get a 25x squared. If I multiply a 5x by a 5x I get a 25x squared. If I multiply 4x squared by negative 1 I get negative 4x squared. If I multiply negative 2x by positive 2x I get a negative 4x squared. So the products of your diagonals are the same. Are exactly the same. They have the same coefficient and they have the same degree. So again you know these patterns are good and we're going to use them when we go to factor. So keep this in mind. Again practice this thing. Look for these patterns each time and hopefully you can master this method. Thank you very much.