The binomial theorem
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Voiceover:It doesn't take long to realize that taking higher and higher powers of binomials can get painful, but let's just work through a few just to realize how quickly they get painful. If we take the binomial a plus b, it's a binomial because it has two terms right over here, let's take that to the 0 power. Anything that's non-zero to the 0 power, that's just going to be equal to 1. That wasn't too bad. Now what about a plus b to the 1st power? That's just going to be a plus b. Now what about a plus b squared? Now, if you haven't been practicing taking powers of binomials, you might have been tempted to say a squared plus b squared, but that would be incorrect. If you did that, you should give yourself a very gentle but not overly discouraging slap on the wrist or the brain or something. a plus b squared is not a squared plus b squared. It is a plus b times a plus b. Then if you do this, it will be a times a, which is a squared, plus a times b, which is ab, plus b times a, which is another ab, plus b times b, which is b squared. You have two ab's here, so you could add them together, so it's equal to a squared plus 2ab plus b squared. Now things are going to get a little bit more interesting. What is a plus b to the 3rd power going to be equal to? I encourage you to pause this video and try to figure that out on your own. Well, we know that a plus b to the 3rd power is just a plus b to the 2nd power times another a plus b. Let's just multiply this times a plus b to figure out what it is. I'll do this. Let's see. Let's multiply that times a plus b. I'm just going to multiply it this way. First, I'll multiply b times all of these things. I'll do it in this green color. b times b squared is b to the 3rd power. b times 2ab is 2a squared, so 2ab squared, and then b times a squared is ba squared, or a squared b, a squared b. I'll multiply b times all of this stuff. Now let's multiply a times all this stuff. a times b squared is ab squared, ab squared. a times 2ab is 2a squared b, 2a squared b, and then a times a squared is a to the 3rd power. Now when we add all of these things together, we get, we get a to the 3rd power plus, let's see, we have 1 a squared b plus another, plus 2 more a squared b's. That's going to be 3a squared b plus 3ab squared. 2ab squared plus another ab squared is going to be 3ab squared plus b to the 3rd power. Just taking some of the 3rd power, this already took us a little reasonable amount of time, and so you can imagine how painful it might get to do something like a plus b to the 4th power, or even worse, if you're trying to find a plus b to the 10th power, or to the 20th power. This would take you all day or maybe even longer than that. It would be incredibly, incredibly painful. That's where the binomial theorem becomes useful. What is the binomial theorem? The binomial theorem tells us, let me write this down, binomial theorem. Binomial theorem, it tells us that if we have a binomial, and I'll just stick with the a plus b for now, if I have, and I'm going to try to color code this a little bit, if I have the binomial a plus b, a plus b, and I'm going to raise it the nth power, I'm going to raise this to the nth power, the binomial theorem tells us that this is going to be equal to, and the notation is going to look a little bit complicated at first, but then we'll work through an actual example, is going to be equal to the sum from k equals 0, k equals 0 to n, this n and this n are the same number, of ... I don't want to ... that's kind of a garish color ... of n choose k, n choose k, and we'll review that in a second; this comes straight out of combinatorics; n choose k times a to the n minus k, n minus k, times b, times b to the k, b to the k power. Now this seems a little bit unwieldy. Let's just review, remind ourselves what n choose k actually means. If we say n choose k, I'll do the same colors, n choose k, we remember from combinatorics this would be equal to n factorial, n factorial over k factorial, over k factorial times n minus k factorial, n minus k factorial, so n minus k minus k factorial, let me color code this, n minus k factorial. Let's try to apply this. Let's just start applying it to the thing that started to intimidate us, say, a plus b to the 4th power. Let's figure out what that's going to be. Let's try this. So a, and I'm going to try to keep it color-coded so you know what's going on, a plus b, although it takes me a little bit more time to keep switching colors, but hopefully it's worth it, a plus b. Let's take that to the 4th power. The binomial theorem tells us this is going to be equal to, and I'm just going to use this exact notation, this is going to be the sum from k equals 0, k equals 0 to 4, to 4 of 4 choose k, 4 choose k, 4 choose ... let me do that k in that purple color, 4 choose k of a to the 4 minus k power, 4 minus k power times b to the k power, b to the k power. Now what is that going to be equal to? Well, let's just actually just do the sum. This is going to be equal to, so we're going to start at k equals 0, so when k equals 0, it's going to be 4 choose 0, 4 choose 0, times a to the 4 minus 0 power, well, that's just going to be a to the 4th power, times b to the 0 power. b to the 0 power is just going to be equal to 1, so we could just put a 1 here if we want to, or we could just leave it like that. This is what we get when k equals 0. Then to that, we're going to add when k equals 1. k equals 1 is going to be, the coefficient is going to be 4 choose 1, and it's going to be times a to the 4 minus 1 power, so a to the 3rd power, and I'll just stick with that color, times b to the k power. Well, now, k is 1b to the 1st power. Then to that, we're going to add, we're going to add 4 choose 2, 4 choose 2 times a to the ... well, now k is 2. 4 minus 2 is 2. a squared. I think you see a pattern here. a to the 4th, a to the 3rd, a squared, and then times b to the k. Well, k is 2 now, so b squared, and you see a pattern again. You could say b to the 0, b to the 1, b squared, and we only have two more terms to add here, plus 4 choose 3, 4 choose 3 times 4 minus 3 is 1, times a, or a to the 1st, I guess we could say, and then b to the 3rd power, times a to the 1st b to the third, and then only one more term, plus 4 choose, 4 choose 4. k is now 4. This is going to be our last term right now. We're getting k goes from 0 all the way to 4, 4 choose 4. a to the 4 minus 4, that's just going to be 1, a to the 0, that's just 1, so we're going to be left with just b to the k power, and b is 4 right over here. We're almost done. We've expanded it out. We just need it figure out what 4 choose 0, 4 choose 1, 4 choose 2, et cetera, et cetera are, so let's figure that out. We could just apply this over and over again. So 4 choose 0, 4 choose 0 is equal to 4 factorial over 0 factorial times 4 minus 0 factorial. That's just going to be 4 factorial again. 0 factorial, at least for these purposes, we are defining to be equal to 1, so this whole thing is going to be equal to 1, so this coefficient is 1. Let's see. Let's keep going here. So 4 choose 1 is going to be 4 factorial over 1 factorial times 4 minus 1 factorial, 4 minus 1 factorial, so 3 factorial. What's this going to be? 1 factorial is just going to be 1. 4 factorial is 4 times 3 times 2 times 1. 3 factorial is 3 times 2 times 1. Let me make that clear. 4 times 3 times 2 times 1 over 3 times 2 times 1 is just going to leave us with 4. This right over here is just going to be 4. Then we need to figure out what 4 choose 2 is. 4 choose 2 is going to be 4 factorial over 2 factorial times what's 4 minus ... this is going to be n minus k, 4 minus 2 over 2 factorial. So what is this going to be? Let me scroll over to the right a little bit. This is going to be 4 times 3 times 2 times 1 over 2 factorial is 2, over 2 times 2. This is 2, this is 2, so 2 times 2 is same thing as 4. We're left with 3 times 2 times 1, which is equal to 6. That's equal to 6. Then what is 4 choose 3? I'll use some space down here. So 4 choose 3, 4 choose 3 is equal to 4 factorial over, over 3 factorial times 4 minus 3 factorial, so that's just going to be 1 factorial. Well, we already figured out what that is. That's the same thing as this right over here. You just swap the 1 factorial and the 3 factorial. We already figured out that this is going to be equal to 4. That is equal to 4. 4 choose 4? Well, this is just going to be, let me just do it over here, 4 choose 4 is 4 factorial over 4 factorial times 0 factorial, which is the exact thing we had here, which we figured out was 1. Just like that, we're done. We were able to figure out what a plus b to the 4th power is. It's 1a to the 4th plus 4a to the 3rd b to the 1st plus 6a squared b squared plus 4ab cubed plus b to the 4th. Actually, let me just write that down, since we did all that work. This is equal to a to the 4th plus, plus 4, plus 4a to the 3rd, a to the 3rd b plus, plus 6, plus 6a squared b squared, a squared b squared, plus, plus, plus 4, I think you see a pattern here, plus 4a times b to the 3rd power plus b to the 4th power, plus b to the 4th power. There is an interesting pattern here. There is a symmetry where you have the coefficient, you go 1, 4, 6 for the middle term, and then you go back to 4, and then you go back to 1. Then you also see that pattern, is that you start at a to the 4th, a to the 3rd, a squared, a, and then you could say there is an a to the 0 here, and then you started b to the 0, which we didn't write it, but that's just 1, then b to the 1st, b squared, b to the 3rd, b to the 4th. This is just one application or one example. In future videos, we'll do more examples of the binomial theorem and also try to understand why it works.