The Binomial Theorem
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You are faced with the problem of expanding .  What to do???  Do you really have to multiply this expression times itself 10 times??   That could take forever.

Let's investigate:

When binomial expressions are expanded, is there any type of pattern developing which might help us expand more quickly?  Consider the following expansions:

What observations can we make in general about the expansion of (a + b)n ?

1.  The expansion is a series (an adding of terms).
2. The number of terms in each expansion is one more than n.  (terms = n + 1)
3. The power of a starts with an and decreases by one in each successive term ending with a0.  The power of b starts with b0 and  increases by one in each successive term ending with bn.
4. The power of b is always one less than the "number" of the term.  The power of a is always n minus the power of b.
5.  The sum of the exponents in each term adds up to n. 
6. The coefficients of the first and last terms are each one.
7. 

The coefficients of the middle terms form an interesting (but perhaps not easily recognized) pattern where each coefficient can be determined from the previous term.  The coefficient is the product of the previous term's coefficient and a's index, divided by the number of that previous term.
         Check it out: 
The second term's coefficient is determined by a4
The third term's coefficient is determined by 4a3b
To Get Coefficient
From the Previous Term:

(This pattern will eventually be expressed as a combination of the form n C k..)

8. Another famous pattern is also developing regarding the coefficients.  If the coefficients are "pulled off" of the terms and arranged, they form a triangle known as Pascal's triangle.  (The use of Pascal's triangle to determine coefficients can become tedious when the expansion is to a large power.)

1
1    1
1    2    1
1    3    3    1
1     4    6     4    1
(notice the symmetry of the triangle)
 

(The two outside edges of the triangle are comprised of ones.  The other terms are each the sum of the two terms immediately above them in the triangle.)


By pulling these observations together with some mathematical syntax, a theorem is formed relating to the expansion of binomial terms:

Binomial Theorem
(or Binomial Expansion Theorem)

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

Our pattern to obtain the coefficient using the previous term (in observation #6), , actually leads to the n C k used in the binomial theorem.
Here is the connection.  Using our coefficient pattern in a general setting, we get:

Let's examine the coefficient of the fourth term, the one in the box.
If we write a combination
n C k  using k = 3, (for the previous term), we see the connection:

 

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


Remember that
and that

 

Examples using the Binomial Theorem:

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

   

The fastest and most accurate way to calculate combinations is to use your graphing calculator.
For calculator help, see the link at the end of the page.

 

2.  Expand  .

Look carefully!! 
This can be a tricky one!

 

The "tricks" involve the use of an expression with an exponent as the "a" value, and also the change of sign between the terms.  For the Binomial Theorem, this problem is actually .

(Let a = (2x4), b = (-y),  n = 3 and substitute.  The parentheses are a "must have"!!!)

Grab your calculator.  Be sure to raise the entire parentheses to the indicated power and watch out for signs.



 

Finding a Particular Term in a Binomial Expansion

What if I need to find just "one" term in a binomial expansion, such as just the 5th term of ?

Let's call the term we are looking for the r th term.  From our observations, we know that the coefficient of this term will be , the power of b will be r - 1 and the power of a will be n minus the power of b.   Putting this information together gives us a formula for the r th term:

The r th term of the expansion of is:

 

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

 

Grab your calculator.

Be sure to include those parentheses!

Be sure to raise the entire parentheses to the indicated power!

 

Check out how to use your TI-83+/84+ graphing calculator with the binomial theorem. Click here.