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.
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 a^{n} and decreases by one in each
successive term ending with a^{0}. The
power of b
starts with b^{0} and increases by one in
each successive term ending with b^{n}. 
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:

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 _{k }(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:
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.
(Let a =
(2x^{4}), 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 5^{th}
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 5^{th}
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
TI83+/84+ graphing calculator with
the binomial theorem.
Click here. 

