Pascal's Triangle, Pascal's Formula, The Binomial Theorem and Binomial Expansions
A binomial expression is the sum, or difference, of two terms. Using Pascal's triangle to expand a binomial expression. 3 . Consider the following examples. The Binomial Theorem and Binomial Expansions. Pascal's Triangle Use this formula and Pascal's Triangle to verify that 5C3 = 5C3 = 3C1 + The binomial expansion of a difference is as easy, just alternate the signs. Explains how to use the Binomial Theorem, and displays the Theorem's relationship to Pascal's Triangle.
Pascals Triangle: How to easily expand binomials using Pascals Triangle
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.
Pascal's triangle and binomial expansion (video) | Khan Academy
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: And then I go down from there.
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.Developing the Binomial Theorem w/ Pascal's Triangle
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: 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? 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. 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: So we have an a, an a.
- Pascal's triangle and binomial expansion
We have a b, and a b. We're going to add these together.
Binomial Theorem and Pascal's Triangle
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?
Pascal's Triangle for expanding Binomials
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?
Each row gives the combinatorial numbers, which are the binomial coefficients. Begin and end each successive row with 1. To construct the intervening numbers, add the two numbers immediately above.
To construct the next row, begin it with 1, and add the two numbers immediately above: Again, add the two numbers immediately above: Finish the row with 1.
There are instances that the expansion of the binomial is so large that the Pascal's Triangle is not advisable to be used. An easier way to expand a binomial raised to a certain power is through the binomial theorem.
It is finding the solution to the problem of the binomial coefficients without actually multiplying out. The theorem is given as: Which can be expanded as: Remember that is another way of writing combination Binomial Theorem also applies to binomial with literal coefficients. It is given as: How to solve different types of problems Proving an expression when terms above a certain threshold can be ignored Question If x is too small so that terms of x3 and higher can be ignored, show that Answer We use the Pascal's Triangle in the expansion of x 6.
The index of x 6 is 6, so we look on the 7th line of the Pascal's Triangle.