Binomial Coefficients and the Binomial Theorem

Binomial Coefficients and the Binomial Theorem

Author: c o

- 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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Before beginning this lesson, you should be familiar with factorials, sequences, and summation notation.


Binomials and Binomial Expansions

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.

Binomial Expansions As Sums

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.


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!

The Binomial Theorem

Binomial Coefficients

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 kth term in the expansion of a binomial that has been raised to the nth 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

The Theorem

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 nCkxn-kyk.

Citing again our example (x+y)3, we see that 


Example 1 - a longer expansion


Example 2 - picking out a coefficient

Bonus - the Pascal's Triangle Connection

This is Pascal's Triangle

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 Connection?

The neat thing about Pascal's triangle is that each line contains the binomial coefficients for a binomial expansion of the nth 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

x4+ 4x3 + 6x2 + 4x + 1

Pretty neat huh?