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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

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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
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
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’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
A Combination

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

Symbolically, it can also be given as


So now what
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


What could these represent
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


Notice anything
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 theorem1
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


Binomial theorem2
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


Practice problems
Practice Problems

  • Evaluate

  • Expand, then evaluate



That seems like a lot of work
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
Example

Given the expansion of

Find

  • The middle term

  • The second term

  • The third term

  • The 9th term


So if you were giving hints
So, if you were giving hints

For the middle term the coefficient is….

why?

For the kth term the coefficient is….

why?


Resources
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 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


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