1. Binomial Theorem

2. 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

3. Do you see anything? 11 (n=0) a + b 1 1 (n=1) a 2 +2ab+b 2 1 2 1 (n=2) a 3 +3a 2 b+3ab 2 +b 3 1 3 3 1 (n=3) a 4 +4a 3 b+6a 2 b 2 +3ab 3 +b 4 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!

4. Lets think a little… When (a+b) 4 was expanded, look at it this way: a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 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.

5. A Combination A Combination n elements, r at a time, is given by the symbol Symbolically, it can also be given as

6. 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

7. 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

8. 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?

9. 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

10. 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

11. Practice Problems 1.Evaluate 2.Expand, then evaluate

12. Practice

13. 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….

14. Example Given the expansion of Find a)The middle term b)The second term c)The third term d)The 9 th term

15. So, if you were giving hints For the middle term the coefficient is…. why? For the k th term the coefficient is…. why?

16. Resources Hubbard, M., Roby, T., (?) Pascal’s Triangle, from Top to Bottom, retrieved 3/1/05 from http://binomial.csuhayward.edu/Pascal0.htmlhttp://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.htmlhttp://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 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 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.htmlhttp://www.krysstal.com/binomial.html Loy, Jim (1999) The Yanghui Triangle, Retrieved 3/1/05 from http://www.jimloy.com/algebra/yanghui.htm http://www.jimloy.com/algebra/yanghui.htm http://mathforum.org/workshops/usi/pascal/pascal_handouts.html