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More examples of special products

Sal gives numerous examples of the two special binomial product forms: perfect squares and the difference of two squares. Created by Sal Khan and CK-12 Foundation.

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Video transcript

And now I want to do a bunch of examples dealing with probably the two most typical types of polynomial multiplication that you'll see, definitely, in algebra. And the first is just squaring a binomial. So if I have x plus 9 squared, I know that your temptation is going to say, oh, isn't that x squared plus 9 squared? And I'll say, no, it isn't. You have to resist every temptation on the planet to do this. It is not x squared plus 9 squared. Remember, x plus 9 squared, this is equal to x plus 9, times x plus 9. This is a multiplication of this binomial times itself. You always need to remember that. It's very tempting to think that it's just x squared plus 9 squared, but no, you have to expand it out. And now that we've expanded it out, we can use some of the skills we learned in the last video to actually multiply it. And just to show you that we can do it in the way that we multiplied the trinomial last time, let's multiply x plus 9, times x plus a magenta 9. And I'm doing it this way just to show you when I'm multiplying by this 9 versus this x. But let's just do it. So we go 9 times 9 is 81. Put it in the constants' place. 9 times x is 9x. Then we have-- go switch to this x term-- we have a yellow x. x times 9x is 9x. Put it in the first degree space. x times x is x squared. And then we add everything up. And we get x squared plus 18x plus 81. So this is equal to x squared plus 18x plus 81. Now you might see a little bit of a pattern here, and I'll actually make the pattern explicit in a second. But when you square a binomial, what happened? You have x squared. You have this x times this x, gives you x squared. You have the 9 times the 9, which is 81. And then you have this term here which is 18x. How did we get that 18x? Well, we multiplied this x times 9 to get 9x, and then we multiplied this 9 times x to get another 9x. And then we added the two right here to get 18x. So in general, whenever you have a squared binomial-- let me do it this way. I'll do it in very general terms. Let's say we have a plus b squared. Let me multiply it this way again, just to give you the hang of it. This is equal to a plus b, times a plus-- I'll do a green b right there. So we have to b times b is b squared. Let's just assume that this is a constant term. I'll put it in the b squared right there. I'm assuming this is constant. So this would be a constant, this would be analogous to our 81. a is a variable that we-- actually let me change that up even better. Let me make this into x plus b squared, and we're assuming b is a constant. So it would be x plus b, times x plus a green b, right there. So assuming b's a constant, b times b is b squared. b times x is bx. And then we'll do the magenta x. x times b is bx. And then x times x is x squared. So when you add everything, you're left with x squared plus 2bx, plus b squared. So what you see is, the end product, what you have when you have x plus b squared, is x squared, plus 2 times the product of x and b, plus b squared. So given that pattern, let's do a bunch more of these. And I'm going to do it the fast way. So 3x minus 7 squared. Let's just remember what I told you. Just don't remember it, in the back of your mind, you should know why it makes sense. If I were to multiply this out, do the distributive property twice, you know you'll get the same answer. So this is going to be equal to 3x squared, plus 2 times 3x, times negative 7. Right? We know that it's 2 times each the product of these terms, plus negative 7 squared. And if we use our product rules here, 3x squared is the same thing as 9x squared. This right here, you're going to have a 2 times a 3, which is 6, times a negative 7, which is negative 42x. And then a negative 7 squared is plus 49. That was the fast way. And just to make sure that I'm not doing something bizarre, let me do it the slow way for you. 3x minus 7, times 3x minus 7. Negative 7 times negative 7 is positive 49. Negative 7 times 3x is negative 21x. 3x times negative 7 is negative 21x. 3x times 3x is 9 x squared. Scroll to the left a little bit. Add everything. You're left with 9x squared, minus 42x, plus 49. So we did indeed get the same answer. Let's do one more, and we'll do it the fast way. So if we have 8x minus 3-- actually, let me do one which has more variables in it. Let's say we had 4x squared plus y squared, and we wanted to square that. Well, same idea. This is going to be equal to this term squared, 4x squared, squared, plus 2 times the product of both terms, 2 times 4x squared times y squared, plus y squared, this term, squared. And what's this going to be equal to? This is going to be equal to 16-- right, 4 squared is 16-- x squared, squared, that's 2 times 2, so it's x to the fourth power. And then plus, 2 times 4 times 1, that's 8x squared y squared. And then y squared, squared, is y to the fourth. Now, we've been dealing with squaring a binomial. The next example I want to show you is when I take the product of a sum and a difference. And this one actually comes out pretty neat. So I'm going to do a very general one for you. Let's just do a plus b, times a minus b. So what's this going to be equal to? This is going to be equal to a times a-- let me make these actually in different colors-- so a minus b, just like that. So it's going to be this green a times this magenta a, a times a, plus, or maybe I should say minus, the green a times this b. I got the minus from right there. And then we're going to have the green b, so plus the green b times the magenta a. I'm just multiplying every term by every term. And then finally minus the green b-- that's where the minus is coming from-- minus the green b times the magenta b. And what is this going to be equal to? This is going to be equal to a squared, and then this is minus ab. This could be rewritten as plus ab, and then we have minus b squared. These right here cancel out, minus ab plus ab, so you're just left with a squared minus b squared. Which is a really neat result because it really simplifies things. So let's use that notion to do some multiplication. So if we say 2x minus 1, times 2x plus 1. Well, these are the same thing. The 2x plus 1, you could view this as, if you like, a plus b, and the 2x minus 1, you can view it as a minus b, where this is a, and that b is 1. This is b. That is a. Just using this pattern that we figured out just now. So what is this going to be equal to? It's going to be a squared, it's going to be 2x squared, minus b squared, minus 1 squared. 2x squared is 4x squared. 1 squared is just 1, so minus 1. So it's going to be 4x squared minus 1. Let's do one more of these, just to really hit the point home. I'll just focus on multiplication right now. If I have 5a minus 2b, and I'm multiplying that times 5a plus 2b. And remember, this only applies when I have at a product of a sum and a difference. That's the only time that I can use this. And I've shown you why. And if you're ever in doubt, just multiply it out. It'll take you a little bit longer. And you'll see the terms canceling out. You can't do this for just any binomial multiplication. You saw that earlier in the video, when we were multiplying, when we were taking squares. So this is going to be, using the pattern, it's going to be 5a squared minus 2b squared, which is equal to 25 a squared minus 4b squared. And, well, I'll leave it there, and I'll see you in the next video.