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Pascal's triangle and binomial expansion

Video transcript
In the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out. And we did it. And it was a little bit tedious but hopefully you appreciated it. It would have been useful if we did even a higher power-- a plus b to the seventh power, a plus b to the eighth power. But what I want to do in this video is show you that there's another way of thinking about it and this would be using "Pascal's Triangle". And if we have time we'll also think about why these two ideas are so closely related. So instead of doing a plus b to the fourth using this traditional binomial theorem-- I guess you could say-- formula right over here, I'm going to calculate it using Pascal's triangle and some of the patterns that we know about the expansion. So once again let me write down what we're trying to calculate. We're trying to calculate a plus b to the fourth power-- I'll just do this in a different color-- to the fourth power. So what I'm going to do is set up Pascal's triangle. So Pascal's triangle-- so we'll start with a one at the top. And one way to think about it is, it's a triangle where if you start it up here, at each level you're really counting the different ways that you can get to the different nodes. So one-- and so I'm going to set up a triangle. So if I start here there's only one way I can get here and there's only one way that I could get there. But now this third level-- if I were to say how many ways can I get here-- well, one way to get here, one way to get here. So there's two ways to get here. One way to get there, one way to get there. The only way I get there is like that, the only way I can get there is like that. But the way I could get here, I could go like this, or I could go like this. And then we could add a fourth level where-- let's see, if I have-- there's only one way to go there but there's three ways to go here. One plus two. How are there three ways? You could go like this, you could go like this, or you could go like that. Same exact logic: there's three ways to get to this point. And then there's only one way to get to that point right over there. And so let's add a fifth level because this was actually what we care about when we think about something to the fourth power. This is essentially zeroth power-- binomial to zeroth power, first power, second power, third power. So let's go to the fourth power. So how many ways are there to get here? Well I just have to go all the way straight down along this left side to get here, so there's only one way. There's four ways to get here. I could go like that, I could go like that, I could go like that, and I can go like that. There's six ways to go here. Three ways to get to this place, three ways to get to this place. So six ways to get to that and, if you have the time, you could figure that out. There's three plus one-- four ways to get here. And then there's one way to get there. And now I'm claiming that these are the coefficients when I'm taking something to the-- if I'm taking something to the zeroth power. This is if I'm taking a binomial to the first power, to the second power. Obviously a binomial to the first power, the coefficients on a and b are just one and one. But when you square it, it would be a squared plus two ab plus b squared. If you take the third power, these are the coefficients-- third power. And to the fourth power, these are the coefficients. So let's write them down. The coefficients, I'm claiming, are going to be one, four, six, four, and one. And how do I know what the powers of a and b are going to be? Well I start a, I start this first term, at the highest power: a to the fourth. And then I go down from there. a to the fourth, a to the third, a squared, a to the first, and I guess I could write a to the zero which of course is just one. And then for the second term I start at the lowest power, at zero. And then b to first, b squared, b to the third power, and then b to the fourth, and then I just add those terms together. And there you have it. I have just figured out the expansion of a plus b to the fourth power. It's exactly what I just wrote down. This term right over here, a to the fourth, that's what this term is. One a to the fourth b to the zero: that's just a to the fourth. This term right over here is equivalent to this term right over there. And so I guess you see that this gave me an equivalent result. Now an interesting question is 'why did this work?' And I encourage you to pause this video and think about it on your own. Well, to realize why it works let's just go to these first levels right over here. If I just were to take a plus b to the second power. a plus b to the second power. This is going to be, we've already seen it, this is going to be a plus b times a plus b so let me just write that down: a plus b times a plus b. So we have an a, an a. We have a b, and a b. We're going to add these together. And then when you multiply it, you have-- so this is going to be equal to a times a. You get a squared. And that's the only way. That's the only way to get an a squared term. There's only one way of getting an a squared term. Then you're going to have plus a times b. So-- plus a times b. And then you're going to have plus this b times that a so that's going to be another a times b. Plus b times b which is b squared. Now this is interesting right over here. How many ways can you get an a squared term? Well there's only one way. You're multiplying this a times that a. There's one way of getting there. Now how many ways are there of getting the b squared term? How many ways are there of getting the b squared term? Well there's only one way. Multiply this b times this b. There's only one way of getting that. But how many ways are there of getting the ab term? The a to the first b to the first term. Well there's two ways. You can multiply this a times that b, or this b times that a. There are-- just hit the point home-- there are two ways, two ways of getting an ab term. And so, when you take the sum of these two you are left with a squared plus two times ab plus b squared. Notice the exact same coefficients: one two one, one two one. Why is that like that? Well there is only one way to get an a squared, there's two ways to get an ab, and there's only one way to get a b squared. If you set it to the third power you'd say okay, there's only one way to get to a to the third power. You just multiply the first a's all together. And there is only one way to get to b to the third power. But there's three ways to get to a squared b. And you could multiply it out, and we did it. We did it all the way back over here. There's three ways to get a squared b. We saw that right over there. And there are three ways to get a b squared. Three ways to get a b squared. And if you sum this up you have the expansion of a plus b to the third power. So hopefully you found that interesting.