![]() ![]() If you set it to the third power you'd say 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. Why is that like that? Well there is only Notice the exact same coefficients: one two one, one two one. And so, when you take the sum of these two you are left with a squared plus Just hit the point home- there are two ways, This a times that b, or this b times that a. Of getting the ab term? The a to the first b to the first term. Of getting the b squared term? Well there's only one way. Of getting the b squared term? How many ways are there ![]() How many ways can you getĪn a squared term? Well there's only one way. Plus this b times that a so that's going to be another a times b. There's only one way of gettingĪn a squared term. ![]() We're going to add these together.Īnd then when you multiply it, you have- so this is going to be equal to a times a. We've already seen it, this is going to beĪ plus b times a plus b so let me just write that down:Ī plus b times a plus b. If I just were to takeĪ plus b to the second power. Go to these first levels right over here. 'why did this work?' And I encourage you to pause this videoĪnd think about it on your own. This term right over here is equivalent to this term right over there. This term right over here,Ī to the fourth, that's what this term is. I have just figured out the expansion of a plus b to the fourth power. 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. 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. 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. The coefficients, I'm claiming,Īre going to be one, four, six, four, and one. If you take the third power, theseĪre the coefficients- third power. But when you square it, it would beĪ squared plus two ab plus b squared. Obviously a binomial to the first power, the coefficients on a and bĪre just one and one. I'm taking something to the zeroth power. These are the coefficients when I'm taking something to the- if There's three plus one-įour ways to get here. Have the time, you could figure that out. Go like that, I could go like that, I could go like that,Īnd I can go like that. Straight down along this left side to get here, so there's only one way. So how many ways are there to get here? Well I just have to go all the way This is essentially zeroth power-īinomial to zeroth power, first power, second power, third power. This was actually what we care about when we think about You could go like this, or you could go like that. How are there three ways? You could go like this, Where- let's see, if I have- there's only one way to go thereīut there's three ways to go here. The only way I can get there is like that. How many ways can I get here- well, one way to get here, But now this third level- if I were to say So if I start here there's only one way I can get here and there's only one way So one- and so I'm going to set upĪ triangle. Up here, at each level you're really counting the different ways So Pascal's triangle- so we'll start with a one at the top.Īnd one way to think about it is, it's a triangle where if you start it We're trying to calculate a plus b to the fourth power- I'll just do this in a different color. Here, I'm going to calculate it using Pascal's triangleĪnd some of the patterns that we know about the expansion. Using this traditional binomial theorem- I guess you could say- formula right over So instead of doing a plus b to the fourth And if we have time we'll also think about why these two ideasĪre so closely related. Of thinking about it and this would be using In this video is show you that there's another way If we did even a higher power- a plus b to the seventh power,Ī plus b to the eighth power. And it wasĪ little bit tedious but hopefully you appreciated it. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in Persia, India, China, Germany, and Italy.To apply the binomial theorem in order to figure out whatĪ plus b to fourth power is in order to expand this out. In mathematics, Pascal's triangle is a triangular array of the binomial coefficients arising in probability theory, combinatorics, and algebra. A diagram showing the first eight rows of Pascal's triangle. ![]()
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