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- to define Binomial Coefficient

- to introduce multimple notations for bionomial coefficient

- to introduce the binomial theorem, and show how it is used for expanding binomials

- to provide examples finding terms in binomial expansions

Binomials and their expansions are first introduced before the learner is presented with the theorem statement. A few examples applying the theorem are then provided. Finally, a short section on Pascal's triangle is given.

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Tutorial

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

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

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

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

Huh? Well, lets say we wanted to calculate * (x+1)^{4}*. Then looking above at the line where

**1 4 6 4 1**

**x ^{4}+ 4x^{3} + 6x^{2} + 4x + 1**

Pretty neat huh?