1. Binomial Theorem

2. 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!

3. 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.

4. 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

5. 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

6. 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?

7. 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

8. 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! I’m a fan of

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

10. Practice

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

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

13. 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