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Binomial Theorem
 

Binomial Theorem
(or Binomial Expansion Theorem)
b1

Most of the syntax used in this theorem should look familiar. The b2notation is just another way of writing a combination such as n C (read "n choose k").
 b3

The Binomial Theorem can also be written in its expanded form as:

b4
Remember that b6 and that b7

 

Example 1:    Expand  b8.
Let a = x, b = 2, n = 5 and substitute.
(Do not substitute a value for k.)

 b9

   b10

b11

Now, grab your graphing calculator to find those combination values.

Method 1:  Use the graphing calculator to evaluate the combinations on the home screen.  Remember:  Enter the top value of the combination FIRST.  Then hit MATH key, arrow right (or left) to PRB heading, and choose #3 nCr.  Now, enter the bottom value of the combination.   

1     2     3

b13

 

Method 2: Use the graphing calculator to evaluate the combinations under the lists. 

4
In L1, enter the values 0 through
the power to which the binomial
is raised, in this case 5.

5
In L2, enter the combination
formula, using the power of the
binomial as the starting value,
and the entries from L1 as the
ending values.

6
The coefficients from the
combinations will appear
in L2.

b14



Finding a Particular Term in a Binomial Expansion
 

The r th term of the expansion of b15 is:

 

Example 2:    Find the 5th term of  b27.
Let r = 5, a = (3x), b = (-4),
n = 12 and substitute. 

 

    b18 
     b19
Grab your calculator.

Be sure to include those parentheses!
(unless you do the subtraction manually)
        7
Be sure to raise each entire parentheses to the indicated powers!

        8