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Binomial Theorem
Task
Let’s experiment and see if you see anything familiar.
Expand these binomials:
If your last name begins with A-F (a+b)0
If your last name begins with G-L (a+b)1
If your last name begins with M-P (a+b)2
If your last name begins with Q-S (a+b)3
If your last name begins with T-Z (a+b)4
Do you see anything?
1 1 (n=0)
a + b 1 1 (n=1)
a2+2ab+b2 1 2 1 (n=2)
a3+3a2b+3ab2+b3 1 3 3 1 (n=3)
a4+4a3b+6a2b2+3ab3+b4 1 4 6 4 1 (n=4)
On the left is the expansion by foiling; on the right is something else… Does anyone recognize it?
Yes! Pascal’s Triangle!
Lets think a little…
When (a+b)4 was expanded, look at it this way:
a4 + 4a3b + 6a2b2 + 4ab3 + b4
There was 1 term that no b’sThere were 4 terms that had one bThere were 6 terms that had two b’sThere were 4 terms that had three b’sThere was 1 terms that had four b’s.
A Combination
A Combination n elements, r at a time, is given by the symbol
Symbolically, it can also be given as
n r
n!C
r!(n r)!
n
r
So now what?
Find the following:If your last name begins with A-F find
If your last name begins with G-L find
If your last name begins with M-P find
If your last name begins with Q-S find
If your last name begins with T-Z find
4
0
4
1
4
2
4
3
4
4
What could these represent?
4 terms, 0 (b’s) at a time
4 terms, 1 (b) at a time
4 terms, 2 (b’s) at a time
4 terms, 3 (b’s) at a time
4 terms, 4 (b’s) at a time
41
0
44
1
46
2
44
3
41
4
Notice anything?
That formula allows you to find all the coefficients for a particular row.
You found the coefficients for the expansion of (a+b)4 power.
Now, what would the coefficients of row 7 be? What do you think would be the easiest way to find it?
Binomial Theorem
This all leads us to Binomial Theorem, which allows you to expand any binomial without foiling. Is it better? Depends on the situation, but it is a good process to understand.
It is all about patterns! Here is The Binomial Theorem
n n n 1 n 2 2 n 3 3
1 n 1 n
n n n n(a b) a a b a b a b ...
0 1 2 3
n n a b b
n 1 n
Binomial TheoremIt looks much worse than it is! Don’t
worry! The key is patterns – if you notice there is a standard pattern for every term!
What do you see? What hints can you give yourself?
I’m a fan of
difference bottomnumberna b
r
Practice Problems
1. Evaluate
2. Expand, then evaluate
9
3
6(3x y)
9...1 9 8 7
3 4 7 843..1 6...1 3 2 1
Practice
6 5 1 4 2 3 3
2 4 1 5 6
6 5 4
6 6 6 6(3x) (3x) ( y) (3x) ( y) (3x) ( y)
0 1 2 3
6 6 6 + (3x) ( y) (3x) ( y) ( y)
4 5 6
1(729x ) (6)(243x )( y) (15)(81x )
2 3 3
2 4 5 6
6 5 4 2 3 3 2 4 5 6
(y ) (20)(27x )( y )
(15)(9x )(y ) 6(3x)( y) 1(y)
729x 1458x y 1215x y 540x y 135x y 18xy y
That seems like a lot of work
And it is….
More likely questions on binomial expansion involve the identification of specific terms of a series.
I’m not going to give you the ways to find it- I want you to think and see what you surmise….
Example
Given the expansion of Finda) The middle termb) The second termc) The third termd) The 9th term
12(x 2y)
So, if you were giving hints
For the middle term the coefficient is….
why?
For the kth term the coefficient is….
why?
n
n2
n
k 1
Resources
• Hubbard, M. , Roby, T., (?) Pascal’s Triangle, from Top to Bottom, retrieved 3/1/05 from http://binomial.csuhayward.edu/Pascal0.html
• O'Connor, J. J. , Robertson, E. F. , (1999) Blaise Pascal. Retrieved 2/26/05 from http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pascal.html
• Weisstein, Eric W. (?) Pascal’s Triangle, Retrieved 2/26/05 from http://mathworld.wolfram.com/PascalsTriangle.html
• Britton, J. (2005) Pascal’s Triangle and its Patterns, Retrieved 3/2/05 http://ccins.camosun.bc.ca/~jbritton/pascal/pascal.html
• Katsiavriades, Kryss, Qureshi, Tallaat. (2004) Pascal’s Triangle, Retrieved 2/26/05 from http://www.krysstal.com/binomial.html
• Loy, Jim (1999) The Yanghui Triangle, Retrieved 3/1/05 from http://www.jimloy.com/algebra/yanghui.htm
• http://mathforum.org/workshops/usi/pascal/pascal_handouts.html
