Polynomial special products: perfect square

- [Instructor] What we're going to do in this video is practice squaring binomials, and this is something that we've done in the past, but we're gonna do it with slightly more involved expressions. But let's just start with a little bit of review. If I were to ask you, what is a plus b squared, what would that be? Pause the video and try to figure it out. Well some of you might immediately know what a binomial like this squared is, but I'll work it out. So this is the same thing as a plus b times a plus b. And then we could multiply this a times that a. So that's going to give us a squared. And then I can multiply that a times that b, and that's going to give us ab. Then I could multiply this b times that a. I could write that as ba or ab, so I'll just write it as ab again. And then I multiply this b times that b, so plus b squared. And what I really just did is apply the distributive property twice. We go into a lot of detail in previous videos. Some people also like to call it the FOIL method. Either way, this should all be a review. If it's not, I encourage you to look at those introductory videos. But this is going to simplify to a squared plus we have an ab and another ab, so you add those together, you get two ab plus b squared. Now why did I go through this review? Well now we can use this idea that a plus b squared is equal to a squared plus two ab plus b squared to tackle things that at least look a little bit more involved. So if I were to ask you, what is five x to the sixth plus four squared, pause this video and try to figure it out. And try to keep this and this in mind. Well there is several ways you could approach this. You could just expand this out the way we just did, or you could recognize this pattern that we just established. That if I have an a plus b and I square it, it's going to be this. And so what you might notice is, the role of a is being played by five x to the sixth right over there, and the role of b is being played by four right over there. So we could say, hey, this is going to be equal to a squared, we have our a squared there. So what is a squared? Well five x to the sixth squared is going to be 25 x to the 12th power. And then it's going to be plus two times a times b. So plus two times five x to the sixth times four. Actually let me just write it out just so we don't confuse ourselves. Two times five x to the, I'll color code it too, two times five x to the sixth times four, times four, plus b squared. So plus four squared, so that's going to be plus 16. And then we can simplify this. So this is going to be equal to 25 x to the 12th. Two times five times four is 40. Two times five is 10 times four is 40. So plus 40 x to the sixth plus 16. Let's do another example. And I'll do this one even a little bit faster, just because we're getting, I think, pretty good at this. So let's say we're trying to determine what three t squared minus seven t to the sixth power squared is. Pause the video and try to figure it out. All right we're going to do it together now. So this is our a, and our b now we should view as negative seven t to the sixth. Because this says plus b, so you could view this as plus negative seven t to the sixth. We could even write that if we wanted. We could write this plus negative seven t to the sixth if it helps us recognize this whole thing as b. So this is going to be equal to a squared, which is nine t to the fourth, plus two times this times this, two times a times b. So two times three t squared is going to be six t squared times negative seven t to the sixth. Actually let me write this out. This is getting a little bit complicated. So this is going to be plus two times three t squared times negative seven t to the sixth power. And then last but not least, we're going to square negative seven t to the sixth. So that's going to be negative seven squared is positive 49, and t to the sixth squared is t to the 12th, t to the 12th power. And so this is going to be equal to nine t to the fourth, and let's see, two times three is six times negative seven is negative 42, and t squared times t to the sixth, we add the exponents, we have the same base, so it's going to be t to the eighth. And then we have plus 49 t to the 12th power. So it looks like we did something really fancy. We have this higher degree polynomial. We were squaring this binomial that has these higher degree terms. But we're really just applying the same idea that we learned many many many videos ago, many many lessons ago, in terms of just squaring binomials.