9.5 The Binomial Theorem

The Binomial Theorem is used when finding the coefficients of expanded binomials.

The Binomial Theorem in the expansion of  :



The coefficient of can be found using:

Another way to write: is.


Example : Find the binomial coefficients of :

Step 1: Plug in to formula=

Step 2: Solve====


Example: Find the binomial coefficients of :

Step 1: Plug into formula =

Step 2: Solve ====


Pascal's Triangle:

Another way to find binomial coefficients is through Pascal's Triangle.

In the triangle, each number is found by adding together the two numbers above it. Example: 3+3=6 These numbers are the same as the coefficients of expanded binomials:

Example: Expand the binomial 

    

    Step 1: Find Coefficients

            The coefficients from the 4^th row of Pascal’s Triangle are 1,4,6,4,1

   

    Step 2: Expand

   

    Step 3: Simplify 

When expanding binomials that have subtraction signs instead of addition you alternate the signs in front of each coefficient.

   Example: Expand the binomial 

   The coefficients are the same but the signs alternate: