Lesson 18.3 : Special Products of Binomials

Something to do, I thought I would start making maybe some videos. This can be sort of a test run to see if this kind of thing is a good way to get the material to you guys. I thought I'd start with a quick well, first off. I hope everyone is safe. I hope that your family is safe. I hope that you are doing what you need to do. I'm here at school in an empty classroom talking to a bunch of tables. So yeah, they're not as interesting as you guys. I kind of wish you were here, but, you know, I understand that you gotta do what you gotta do. I gotta do what I gotta do. So I thought we would create some videos in case you guys wanted to get ready for next year and get ready for your continuing education and math. This video is just gonna be a review basically of what the last thing I taught you before we went on break the extra credit problems on the quiz that you guys took. Is what we're gonna be going over in more detail with some practice problems if you guys want to work on it. So yeah, so why don't you, if you have your book turn to page 8, 67, if not, I will upload a PDF of the book with. This section basically. The section of the book will be uploaded as a PDF with this stuff. So you guys can have access to that if you don't have your book with you and don't want to go outside to get it. So anyway, we're going to be multiplying a special where no special products to binomial. So this is like of the form like something some binomial. So here we got a binomial. And we're going to be squaring that binomial. So that's what these special products are on page 8. 67. If you guys remember when we first kind of did these things, or we first kind of ended up multiplying binomials binomials, we did this these tiles. So I think these tiles are good for students that sort of have a difficulty with the abstract nature of this kind of problem and want to see some more like physical things. So we'll start off using the tiles to how to do this just like we did with the other binomials and then we'll go to sort of the more abstract pure math version of it. Now, I think that I saw at least when I was grading your quizzes, I saw a lot of you when you were doing this problem, you thought maybe you could just kind of like factor in that square. That's really not how squares work. Just if we go back to sort of basics, if we have X squared, that's the same thing as X times X so if I have. Two X plus three, and I'm squaring that, that's the same thing as two X plus three times two X plus three. So this is actually what we'll be doing. This is actually what this thing equals. It doesn't, you can't kind of factor in or something like that. Don't do that. You're going to have to do it this way. So let's see how that works using tiles. So just to remind you, I put the key here. Just to remind you what these tiles represent. So if you have a square X, then it's represented by a positive square, if you have a negative square X, it's going to be represented by a negative square. And then you have an X is going to be represented by kind of like a line, and then a negative X would be a negative line and the numbers are going to be dots. So if you have two X's, we got two X's over here. If you have two X's, then we'll represent that by one line two lines. So those are two X's. And then we have three numbers, or a three. So that's one plus one plus one. So that's three. And so we have two X's. And one three, and then we're going to square it. So we got two X's over here. And then we have our three numbers here. And so we're going to be multiplying this X by this X, and when we multiply an X by X, we get a square X, and then we're going to multiply this X by this X, and so we multiply an X by an X, we get another square X, and then we're going to multiply this X by this number. And whenever you multiply anything by one, it equals itself. So I would get an X times one is going to be X and X times one is going to be X and then X times one is going to be X so if you multiply three by X, you get three X's. So again, down here, we're going to do X times X so X times X is going to be X squared, X times X is going to be X squared. And then we're going to get X times three, and so if I multiply X times three, I'm going to get three X's. And I don't really want to do these. Let's see if I can cheat. Yeah, all right, so down here is probably easy to read in my handwriting anyway. But down here, if I do an X, a one times an X, I get an X, a one times an X, I get an X, a one times a one. I get a one, a one times a one. I get a one, a one times a one. I get a one. And then repeat that for this row. So one times X is X, one times X is X, one times one is one, one, one, one, or one X, one X, and then one, one, one. So I've now multiply. I've now multiplied two X plus three, and I multiply that by two X plus three. And so when I've done that, this is what I end up with. And so if I were to figure out what I got, I have four square X's, so four, X squared, I have one, two, three, four, 5, 6, and then another one, two, three, four, 5, 6, I have another, I have 12, X's, and then finally, I got 9. Ones. And so if I were to bring this out, that means that this is what two X plus three squared is. This is what two X plus three times two X plus three would equal. Equal four X squared plus 12 X plus 9. And I'm sorry that I have to do this all in one take because I don't know how to edit videos at all, so I'm gonna have to get some water or something as I go through this. Okay, so just to do two more of these kind of problems and then this is also going to be a special function and I'll show you why it's considered special a second where I'm page 8 67. I've got a key there I can't really see it. Let's move it up here and you won't build it. Okay. Well, here's the key just to remind you X squared is a square X negative X is a negative squared. X is a line, negative X is a negative line, and then numbers are dots. So if I wanted to square if I wanted to square two X plus minus three, and I'm squaring this, so again, this is just going to be two X minus three, and then multiply by two X minus three. And so I got two X's and I got three negative dots. I've got two X's and I've got three negative dots. So I'm going to do the same thing. I won't burden you with my bad handwriting, but I'm going to multiply this X by that X and then I'm going to get a square X and then I'm going to multiply this X by that X, I'm going to go square X, I'm going to multiply that X by that one, and so anything multiplied by one is itself. Anything multiplied by negative one is it's negative self. So X times negative one, oh my gosh, the teachers are causing issues in the hallway without any students. They're just having a party. Let me shut the door. Sorry, then interruption. The other day they're just they're just animals out there. The teachers with no students. It's really it's really anyway. Anyway, back to the map. Go back to the map. Guys stay focused and these tough times. So we have an X times a negative one. That equals a negative X and X times a negative one that equals a negative X and then an X times a negative one will equal another negative X X times X is X squared, X times X is X squared, X times negative one is negative one, negative one. Negative one. Negative one times X is still negative one. Negative one, and then a negative times a negative is the problem with the room being empty. The room, if you guys are here, I know you would have said and I hope I hope you said it home and negative time for negative is a positive. It is a positive. So if I have a negative one times a negative one, it goes positive one, positive one, positive one, negative times negative X is negative, I'm sorry. A negative one times X is a negative X negative X positive one positive one positive one. Negative X, negative X, positive one, positive one, positive one. So now we've multiplied. These numbers together. Are these binomials together? And so we have one, two, three, four. Square X's. We've got 6 negative X's over here. 6 negative X's over here, so that's 12 negative X's. And then finally we have 9 dots. And so that's a 9. So this is what we're ending up with. And you hopefully can see, like this is not the same thing as just like squaring that in there. This I mean, this number is the same. This number is the same if you remember to square the negative two, but this extra term is here. This is the one that everyone forgets. So don't forget that extra term. So and again, this is the same thing as saying two X minus three times two X minus three. Now this is the last, okay, so pause to see if you guys are here, you would maybe pause the best question, maybe. Any questions any questions if you share this? No, okay. Okay. So finally, we've got C and so this is also known as a special product, and I'll show you why in a second, this is actually this actually is going to get an answer that you might have expected this to get. So you might have expected this to get the answer that this is actually going to get. So this is not squaring anything. We're multiplying two X plus three times two X minus three. And so if I were to do that, I would have my this is my two X plus three on the top over here. And this is my two X minus three over here. And so I got to X's. I have negative three, two X's and I have a positive three. So let's see what happens when I do this one. So next time X to X squared, then X times X is an X squared. And X times a one is itself. And X times a one is itself an X times a one is itself. X times an X is an X squared. X times X is X squared. X times one is itself. X times one is itself. X times one is itself. Negative one times X is negative X negative one times X is negative X and then a negative times a positive is a negative. So these are all negative, supposedly. You can see that thing I should have maybe done this one too. Oh man, someone's coming in. Hey Doc. What's up, man? I'm just teaching on the web. Mister Mike said hi to my class. Bye. Bye y'all doing. It's pretty easy. He's regulating, you gotta regulate the classroom. No one here. And all right, I'm sorry. I don't know how the problem is I don't know how to edit videos, so I gotta just keep growing, you know? Like I have to, I don't know how to stop and start it again, so. It's good though. It's like a real, real class environment now. Yep. You know, you just gotta get some kids to throw spitballs or something. They do that. Hold on, I'll find something. There you go. There you go. You no longer will have recycling. Oh no. You ever got all the papers that no one's going to generate trash. Okay. All right. All right. So negative times a positive is a negative, a negative times a positive is a negative. A negative times a positive is still a negative, and negative times a positive is still a negative and negative times a positive is a negative. A negative times a positive is a negative. I'm going to keep saying this. 9 times a positive is negative. You guys remember that. And remember that, right? Negative times a positive is a negative. Negative, negative, negative. Now, those of you that are paying attention might notice that this, and this are reverse. And so these are all positives up on top. And these are all negatives down on bottom, and there are 6 positives. And there are 6 negatives. So the total number of X's I have is going to be zero. They're not going to be any exes. Are you ready to cancel each other out? Now I do have some square X's over here. They're not going to cancel out. We got a four. And then we do have some negative ones over here, and so that's a 9. So the answer to this problem, the answer to two X plus three times two X minus three is going to be four X squared -9. And so if this looks like what you might expect to have gotten over here, but this over here does not get this answer. This over here gets this answer. This is what gets you this answer. So that's kind of why it's considered a special product too because you do a special answer. You end up getting rid of this middle term because these terms will always cancel out with those terms in this sort of situation for these sort of special special products of binomials. So just to go through, oh, yeah, so you might have noticed there is sort of a pattern here. So the pattern is that if you have an a plus a B and you're squaring it, and again, I just want to emphasize that a plus B squared does equal a plus B times a plus B and so I would write this down when I'm doing these kind of problems. I would when I see this problem, I would rewrite it as this kind of problem. But there is a pattern for this kind of problem. That's why they call it a special polynomial. Or a special, you know, that's why these are considered special. So if you take this and you actually do it out and you actually do 8 times a, B times B, B times a, B times B, you're going to get a squared plus two AB, so there's going to be two sets of there's going to be one set of AB here. And then you're going to have another set of AB here, and so you're going to end up with two AB, and then you'll have a B squared. So this works for any set of things. Any binomial being squared will get us that answer. So just to show that, if we have X plus four, and we assume that X is our a and four is our B, then we should get this result. We should get a squared plus two AB plus B squared. And so this is sort of the equation. And so now we're going to do it. We'll have X plus four squared. And so if I have an I would, again, I would just rewrite this as X plus four times X plus four. And then I would do X times X, which is X squared. I would do X times four, which is X was four X, and then I would do four times X, which is four X, and then I would do four times four, which is four squared. And so I have an X squared. I have two sets. Of X times four, and then I have a four squared. And so when I do that out, I get this value, and this value is two times AB. So this would be two times AB, a squared, and B squared. So it would be, yeah. So it's like it's this only with different numbers. And so if I do it out over here, so again, for maybe I should also say pause, are there any questions empty classroom with the door shut that is now locked? So mister Mike can't steal any more of my recycling. Okay, so we're going to do this. So again, remember that maybe you should just rewrite it. If you're having trouble with these kind of problems, just rewriting them is what they actually equal and then going from there is probably maybe a better strategy. What we're going to do three X times three X and then three X times two Y and then we're going to do a two Y times three X and then two Y times two Y so we're going to end up with two pairs of three X times two Y and two pairs of so there's one pair and there's another pair. And so to make it more abstract enough in case you hadn't noticed we're not using the tiles anymore so now this is sort of the more abstract version of it. But if we're like this is now is now our three X and two Y's are B and so we should get it of this form. So let's see if we do that. So we get three X squared. We have three X times three X we have two three X's times two Y's. So we got two of those. And then we have an X two Y squared. So this is a two Y squared. Now if you square a three X squared, you'll get a 9 X squared. If I multiply all these numbers together, so two times three times two is going to be 12. And then two Y squared is going to be Y squared, and then four. So remember that when you have something like this, when you have. Two Y squared, you got to square the two and you got to square the Y so you're going to be left with four Y squared. So just try to keep that in mind when you're doing this. So here you can factor in the two because this is all being multiplied by itself. So like two, two Y quantity squared is equal to two Y or two times Y times two times Y and so that's why you can get this result. If you had two plus Y squared, that would equal two plus Y times two plus Y so then that's why you have to go through all the rigmarole of doing four plus two Y and then plus two Y. Plus Y squared. And so our final answer in standard form would be Y squared plus four Y plus four. So that's how you do these square problems. And so these problems, you can basically just distribute in that two squared, but these problems you can't because of just the way that. What the square means because of what the square mean. So hopefully you could hear all that. I was sort of mumbling to myself down on the floor. Again, I can't really edit this video. So, but let's just work a couple of problems. So these are going to be practice problems. For those of you that want to be able to do some math moving forward and want to be able to continue your math education, I would recommend going to page 8 72 and just looking at these kind of problems. Now if you're super good and I can remember that formula we went over, you can use that. And if you can't, I would just rewrite what the heck X plus H squared actually equals. And I would use whatever method I'm most comfortable with, doing these kind of problems. So X times X is X squared. X times 8 is 8 X, and then I have to do the other set. So for each one of these, you're out to multiply four times because you've got four, you got four terms total. So 8 times X is 8 X and then 8 times 8. Oh my gosh, no one's in the room. Wait, okay, I think it's my hand just kind of rode 40 64 by itself. I don't know if that's actually true or not. 8 times 8 is 64. Yeah, yeah. Like the Nintendo game. Okay, all right. So then we want to simplify this. So these two terms go together. Another color for that. These two terms go together. I get 16 X and then there are no other terms to combine with that last term. So that is where I leave it. All right, pause any questions, you guys give it. All right. So number two, again, I would just rewrite what this equals. So four X plus 6 Y times four X plus 6 Y so four times four X is going to be 16 X squared. And then four X times 6 Y is going to be 24 XY. And then 6 X times four Y is going to be 20 four X, Y, and then 6 X times 6 Y is going to be 30 6 Y squared. So I then combine like terms, so I get 12 X squared plus 48 X, Y, because these two terms are the same so I can combine them. And then 36 Y squared, so then that is our answer. So that's how you do these kind of problems. Now, for these problems where you have some term plus some other term and then you have that same term again minus the same term before. Those are going to simplify to just this is like kind of what you'd expect if you're doing a square. You would do a squared minus B squared is going to be your final answer. Now the reason that is the final answer is not because it's some kind of magic, but because if you actually work through it, you'll see that that is the answer. So here we have our a is X and our B is 6. And if I was to go through and do it, if I was to do X times X, I would get X squared plus two X times negative 6, I get negative 6 X if I did negative or positive 6 times X, I get positive 6 X and if I did 6 times negative 6, I get negative three, 6. And so if I do that, I would end up with these two terms canceling. So there's two middle terms here with cancel. And I would be left with X squared -36. And so this works no matter how complicated we get, we get it. It's still going to work. So if I do X times X squared or X squared times X squared, that's X to the fourth. X squared times. Two Y is going to be X squared or sorry, X squared minus two Y is going to be negative X squared Y and then if I do positive two Y times positive X squared, I'm going to get positive two X squared Y and then I'm left with a negative four Y squared down here at the end. So. These two middle terms then will cancel. So these two middle terms here are going to cancel. And so I'm going to be left with X squared squared, and I'm going to be left with negative four Y squared. And so X squared squared is positive as one four X X to the fourth, and then four Y squared is going to be four squared, which is 16. And then Y squared squared, which would be Y to the fourth. So on the practice, I'm going to do X times X so this is on page 8. 73 jump we're jumping to the practice. So for these two kind of problems, I'm going to have an X times X so that's going to be an X squared. I'm going to have an X times a negative four, so that's negative four X, and then I have a four times a X, so that's going to be four X and I'm going to have a negative four times a positive four. So that's going to be a negative 16. And so these two terms in the middle here cancel. So I'm left with X squared minus 16. Same thing over here. So I'm going to have an X squared times an X squared. So it's X to the fourth. And X squared times a negative 6 Y so it's going to be negative 6, Y X squared. A 6 Y times a positive X squared is going to be a positive 6 Y X squared, and then a 6 Y times a negative 6 Y is going to be negative 36 Y squared. And then these two middle terms drop out. And I'm left with X to the fourth. -36. Y squared. And just because I want to do a problem to show you guys why this sort of stuff can be applied to the real world. So I know it seems kind of abstract and all that jazz, but there are real world problems that involve multiplying two binomials by each other. I'm going to say you have a flower bed. You want to make a flower garden and you want to have a let's say you want to have a patio. And you want to have two things inside there. You're going to have a square patio. And you want to surround the square patio with the flower yard. So you want to have square patio in the center. And you want to have a flower garden around the outside. So the square patio, you're not quite sure, you're not quite sure how much that square patty is going to be. But you do know it's going to be how many tiles you can buy or whatever minus three. And it's going to be square. So one side is going to equal the other side. And then you're going to want to have, let's see, the surrounded by a flower garden with a uniform with the length of a side of the entire square area, including the patio and flower garden, write the expression to the area of the flower garden. So I want to have a patio. So this is my patio. And then I want to have like a flower garden around it. And I want this flower garden. We'll see if I can draw it a little better. So you got a patio here. And this patio is square, and the two sides, I'm not quite sure what X is going to be because I haven't worked that out. Maybe there's some reason that I don't know what actually is going to be yet, but I do know that it's going to be X minus three. And then around it, I want to have a flower garden. And that flower garden is going to be X plus three. So maybe three is like how many feet between the two things I want or something like that, and I don't know quite how big I want the patio to be yet, because I don't know what quite a big patio is going to be yet. I don't know how big the flower garden is going to be yet, but I want to get sort of a sense. I want to have an equation that will give me a sense of the size of this thing. So this is what the two pieces are. We have a patio in the center, and then we have flower flowers around the outside. Some flowers around the outside of my patio. And I want to know how about, you know, I want some kind of way of measuring how many flowers I'm going to have. I want to write an expression for the area of this flower garden, even though I don't have complete information yet. But what I do know is I do know that the area of the patio is going to be X minus three times X minus three because that's how area works. With times Lang, and so that's going to be X squared, and then so X times X is X squared, X times negative three is negative three X so X times X is X squared, X times negative three is negative three X and then I'm going to have negative three times X so that's negative three X and then negative three times negative three that's positive 9. So my area of my. Of my the area of my patio is going to be X squared -6 X plus 9. And so once I have a sense of X is like maybe that's the number of tiles I can afford or that kind of thing. Then I'll have a sense I can just plug it in here and I can get my answer for the patio. The entire thing, the entire the entire outside length of this thing is going to be X plus three times X plus three. So the entire length of this thing is going to be X plus three. So it's bigger than the patio. And then X plus three. So I'm going to do X times X, which is X squared, X times three, which is positive three X, and then I'm going to do three times X so that's three X and I'm going to do three times three, which is 9. And so the entire length of this patio, and flower garden is going to be X squared plus 6 X plus 9. And so now the question asks me about the flower guard. So if I have a patio in the center. And I have my flower garden around the outside, not including the patio. I can take the expression for the one and subtract it from the expression from the other. So I'm going to take the top expression or the bottom expression because that's the bigger one. And I'm going to cut out the expression for the patio. So if I wanted to do this, I would distribute the negative, so whenever you're given two things, and this is a good problem because it's combining the two things that we learned before break. Got to distribute the negative. So this term cancels with this term. This term combines with that term to get 12 versus 12, 16 X 12 X, and then this term cancels with that term. So I'm left with 12 X and so that's how you could do these types of problems. Might not be a bad idea to draw yourself a little picture. And then once you have a little picture that might help you figure out what you're subtracting from what and what the problem is asking. So these ones are certainly certainly a stretch just to do one more that would be on the practice. We have a square swimming pool surrounded by a cement walkway with a uniform width. And so again, we got squares here, so that's why you would be squaring the sides. So the swimming pool is a length of this. And a side length of the entire square pools is that. Write an expression for the area of the walkway. So we have a we have a pool, a square pool, and so this side is going to be X minus two, and since it's a square, that means that this side is also going to be X minus two, and then around it. We're going to have some kind of square walkway. It's not really as well as the main little square ish. All right, all right. All right, my little square here. So this would be X okay. So then this square walkway would have X plus one X plus one. And so then I want to know what the area of the square walkway is. Well, I'll take the area of the pool, which is X minus two times X minus two, and so then I'll get X times X, which is X squared, X times negative two, which is negative two X, and then I'll get negative two times X which is negative. Two X, and then I'll have negative two times negative two, which is a positive for. So that's the area. So once I combine this, so I would get X squared minus four X plus four. So that's the area of the pool. And now I'm going to do the area of the walkway. So the area of the walkway would be X plus one times X plus one. So the length times the width, and so I do X times X, which is X squared. I do X times one, anything times one is itself, so that's X one times X is X, and then one times one is one. And so the area of the whole thing will be this. Now I want to know the area of the walkway. So this is the area of the thing inside the walkway. And this is the area of the whole thing. So if I subtract them, if I take the bigger thing, the area of the whole thing being the bigger thing, and I subtract the area of the pool, that will get me the area of the walk. So those two things attracted will get me the area of the walkway around the pool. And so I wanted to shoot with that negative when I remember I'm subtracting a two polynomials. I always wanted to distribute the negative. So I get negative X squared, and negative times a negative is still a positive, so I get a positive four X, and then I get a negative four. So here we don't have as many cancellations as before, but this cancels with that. So I'm left with no square X's. This combines with that, so I'm going to have to 6 X's and then negative one or positive one minus four is negative three. And so that is an expression for the walkway. And then if I want to do with X being 7, so that's the final thing it asks me to do. Is what effect equals 7? You know, what if what if X equals 7, so it would be 6 times 7 minus three. So 6 times 7 is romantic in the 7. They're my weakness. 6 times 7 would be 24 or sorry, 42. 42 minus three then would be 39. So that would be the square footage of that. Patio. Anyway, so that's where that walkway. So that's the problems. I would really appreciate it if you did a couple of practice problems just to make sure that you can do it. It's a good chance to check to see if you actually learn anything from this from this video. And I guess I'll see you guys when I see you. I will be putting up some solutions to select problems also with these, I believe. So you can check your answers to make sure you're getting it correctly. Have a good day guys. I hope the classroom will eventually have students in it. The chairs are giving me funny looks at this point. So anyway, I hope you guys stay safe and I hope I hope yeah, I hope to see you soon. Talk to you then.