Polynomial special products: difference of squares

- [Instructor] Earlier in our mathematical adventures, we had expanded things like x plus y times x minus y. Just as a but of review, this is going to be equal to x times x, which is x squared; plus x times negative y, which is negative xy; plus y times x, which is plus xy; and then minus y times y. Or you could say y times a negative y, so it's going to be minus y squared. Negative xy, positive xy, so this is just going to simplify to x squared minus y squared. And this is all review. We covered it, and when we thought about factoring things that are differences of squares, we thought about this when we were first learning to multiply binomials. And what we're going to do now is essentially just do the same thing, but do it with slightly more complicated expressions. And so, another way of expressing what we just did is we could also write something like a plus b times a minus b is going to be equal to what? Well, it's going to be equal to a squared minus b squared. The only difference between what I did up here and what I did over here is instead of an x, I wrote an a; and instead of a y, I wrote a b. So, given that, let's see if we can expand and then combine like terms for, if I'm multiplying these two expressions. Say I'm multiplying three plus 5x to the fourth times three minus 5x to the fourth. Pause this video, and see if you can work this out. Alright, well, there's two ways to approach it. You could just approach it exactly the way that I approached it up here, but we already know that when we have this pattern where we have something plus something times that same original something minus the other something, well that's going to be of the form of this thing squared minus this thing squared. And remember, the only reason why I'm applying that is I have a three right over here and here, so the three is playing the role of the a. So, let me write that down, that is our a. And then the role of the b is being played by 5x to the fourth. So, that is our b right over there. So, this is going to be equal to a squared minus b squared. But our a is three, so it's going to be equal to three squared, minus, and then our b is 5x to the fourth, minus 5x to the fourth squared. Now, what does all of this simplify to? Well, this is going to be equal to, three squared is nine, and then minus 5x to the fourth squared. Let's see, 5 squared is 25. And then x to the fourth squared, well, that is just going to be x to the fourth times x to the fourth, which is just x to the eighth. Another way to think about it are exponent properties. This is the same thing as 5 squared times x to the fourth squared. If I raise something to an exponent and then raise that to another exponent, I multiply the exponents. And there you have it. Let's do another example. Let's say that I were to ask you, what is 3y squared plus 2y to the fifth times 3y squared minus 2y to the fifth? Pause this video, and see if you can work that out. Well, we're going to do it the same way. You could, of course, always just try to expand it out the way we did originally. But we could recognize here that, hey, I have an a plus a b times the a minus a b. So, that's going to be equal to our a squared. So, what's 3y squared? Well, that's going to be 9y to the fourth minus our b squared. Well, what's 2y to the fifth squared? Well, 2 squared is four, and y to the fifth squared is y to the five times two, y to the 10th power. And there's no further simplification that I could do here. I can't combine any like terms. And so, we are done here as well.