Polynomial special products: perfect square | Algebra 2 | Khan Academy

What we're going to do in this video is practice squaring binomials. This is something that we've done in the past, but we're going to 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 + 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 + b) times (a + b). Then we can multiply this a times that a, so that's going to give us a squared. Next, 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. Finally, I multiply this b times that b, so plus b squared.

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 2ab plus b squared.

Now, why did I go through this review? Well, now we can use this idea that (a + b) squared is equal to a squared plus 2ab plus b squared to tackle things that at least look a little bit more involved. So if I were to ask you, what is (5x^6 + 4) squared, pause this video and try to figure it out and try to keep this and this in mind.

Well, there's 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. If I have (a + b) and I squared, it's going to be this. What you might notice is the role of a is being played by 5x^6 right over there, and the role of b is being played by 4 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, (5x^6) squared is going to be 25x^12. Then it's going to be plus 2 times a times b, so plus 2 times 5x^6 times 4. Actually, let me just write it out just so we don't confuse ourselves: 2 times 5x^6 times 4. Plus b squared, so plus 4 squared, so that's going to be plus 16.

Then we can simplify this. So this is going to be equal to 25x^12. 2 times 5 times 4 is 40. 2 times 5 is 10, times 4 is 40, so plus 40x^6 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 (3t^2 - 7t^6) 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 b as -7t^6 because this says plus b. So you could view this as plus (-7t^6). We could even write that if we want; we could write this plus (-7t^6) if it helps us recognize this whole thing is b.

So this is going to be equal to a squared, which is 9t^4, plus 2 times this times this. 2 times a times b, so 2 times 3t^2 is going to be 6t^2 times -7t^6. Actually, let me write this out; this is getting a little bit complicated. So this is going to be plus 2 times 3t^2 times -7t^6.

Then last but not least, we're going to square -7t^6, so that's going to be -7 squared, which is positive 49, and t^6 squared is t^12. So this is going to be equal to 9t^4.

Let's see, 2 times 3 is 6 times -7 is -42, and t^2 times t^6, we add the exponents, and we have the same base, so it’s going to be t^8. Then we have plus 49t^12.

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.