Before beginning this lesson, you should be familiar with factorials, sequences, and summation notation.
A binomial is a polynomial with two terms. For example x + 1 and y + x and 2x + 3z are all binomials. When we calculate an expression like (x+1)^{2} or (y+x)^{3} we end up with a binomial expansion. Finding binomial expansions can become too tedious to do by hand when dealing with large exponents. Just look at these examples
Already with an exponent of 3 the algebra starts to get a little obnoxious.
Notice that in both of the above binomial expansions the resulting polynomial is just a sum of terms. In this case, they can be rewritten using summation notation.
and
Using summation notation, we can take a long and potentially confusing expansion and make it a little more concise. It would be nice, however, if there were some way to calculate the coefficients of the binomial expansions. Enter the binomial theorem!
Before we get to the theorem, we should understand its crucial component, the binomial coefficient. Here is the notation and its definition:
This notation is usually pronounced as "n choose k", and it represents the coefficient of the k^{th} term in the expansion of a binomial that has been raised to the n^{th} power. For instance, in the above example of (x+y)^{3}, the second binomial coefficient (when k = 2) is 3. Alternative notations for the same thing are
We are now ready to present the binomial theorem in its unvarnished glory:
This states the the binomial expansion of an expression like (x+y)^{n} is a sum of terms of the sequence whose general term is defined to be _{n}C_{k}x^{n-k}y^{k}.
Citing again our example (x+y)^{3}, we see that
And so on and so forth, you can extend Pascal's triangle on into infinity. The basic procedure is to first place a 1 along the edge, then each term in the row is the sum of the two terms it lies between on the preceding row. The arrows should make its pattern pretty clear.
The neat thing about Pascal's triangle is that each line contains the binomial coefficients for a binomial expansion of the n^{th} degree, where n is the line number!
Huh? Well, lets say we wanted to calculate (x+1)^{4}. Then looking above at the line where n=4, we can just write
1 4 6 4 1
x^{4}+ 4x^{3} + 6x^{2} + 4x + 1
Pretty neat huh?
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