Binomial Theorem - Pascal's Triangle - An Introduction to Expanding Binomial Series and Sequences.
Binomial expansions using Pascal's triangle and factorial notation. Consider the following expanded powers of (a + b)n, where a + b is any binomial and n is a whole number. How can we determine the value of each coefficient, ci?. Pascal's Triangle gives us the coefficients for an expanded binomial of the To find the rth term of a binomial expansion (a + b)n, plug the terms into the formula. A binomial expression is the sum, or difference, of two terms. For example, x + 1, 3x Pascal's triangle. 2. 3. Using Pascal's triangle to expand a binomial expression .. Find the coefficient of x5 in the expansion of (1 + 4x)9. 6. In the expansion.
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. How are there three ways? You could go like this, you could go like this, or you could go like that.
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.
Pascal's triangle and binomial expansion (video) | Khan Academy
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. 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.Using binomial expansion to expand a binomial to the fourth degree
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. 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. Note that each term is a combination of a and b and the sum of the exponents are equal to 3 for each terms.
The degree of each term is 3. Thus, if we expand the question is, what are the coefficients? These coefficients are called binomial coefficients. Note that there is a "left to right" and "right to left" symmetry to the numbers. Theses coefficients can be obtained by the use of Pascal's Triangle.
Pascal's Triangle This triangular array is called Pascal's Triangle. 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.