9.5: The Binomial Theorem

Hello folks, let's learn about the Binomial Theorem....

So a binomial is just a polynomial that has two terms, like so:



Where n is any number.  The Binomial Theorem helps you calculate binomial coefficients; it provides a quick way to raise a power of a binomial.

Let's find some coefficients!

When you expand

 , you get this:



Each term has three parts; the binomial coefficient, the x value, and the y value.  We'll bother about the x and y values of each term in a second, but first here's how to find the coefficient of each term, or nCr.

The coefficient of             is:




....nCr can also be written as :

 

Here's an example:
 


**In general, it is true that    

Pascal's Triangle

Pascal's Triangle is a VERY convenient way to remember a pattern of coefficients. 


The first and last number of each row is 1, and every other number in each row is formed by adding the two numbers above the number:

 

Pascal's Triangle shows the coefficients for each term in the expansion of a binomial...

The top row of Pascal's Triangle is called the zero row, and it is the expansion of the binomial


The next row is called the first row; it is the expansion of


Following suit, the next row is called the second row, and it shows the coefficients of each term in the expansion of


In general, the nth row in Pascal's Triangle gives the coefficients of    

X and Y values of each term in the binomial expansion

To find the x values in each term, you start out with , then go  for the second term, and  for the third term.  The expansion always ends with , which is one.

To find the y values in each term, start with , which is one.  The next term's y value is , and the third term is .  Keep adding one to the power when you find the next term, and the final term should be .

It's quite simple, really!!

    
Let's try one:

write the expansion for the expression

:

The binomial coefficients from the fourth row of Pascal's Triangle are 1, 4, 6, 4, 1. 

--For expansions of binomials representing differences, rather than sums, you alternate signs--

Therefore,



Now, simplify:




And there you have it boys and girls! The Binomial Theorem! 

Henry