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Binomial Theorem

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Binomial Theorem

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

11 (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!

When (a+b)4 was expanded, look at it this way:

a4 + 4a3b + 6a2b2 + 4ab3 + b4

There was 1 term that no b’s

There were 4 terms that had one b

There were 6 terms that had two b’s

There were 4 terms that had three b’s

There was 1 terms that had four b’s.

A Combination n elements, r at a time, is given by the symbol

Symbolically, it can also be given as

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 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

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?

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

It 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

- Evaluate
- Expand, then evaluate

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….

Given the expansion of

Find

- The middle term
- The second term
- The third term
- The 9th term

For the middle term the coefficient is….

why?

For the kth term the coefficient is….

why?

- 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 fromhttp://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