It is about time to have a project and exam. Your project will be one of the three outlined below. Whichever one you choose, it will be submitted as a maple worksheet, with text and formatted inline mathematics. Calculations with Maple are encouraged and I will positively beam if you define a word or two by a Maple procedure. Your project will be individually prepared and submitted by class time Tuesday 10/5.

1. Show that the set of polynomials  where  satisfy the identity (***)  . In semigroup parlance, this says that the function  is an isomorphism between the semigroup  of all polynomials  under composition and the semigroup  of non-negative integers under mulitplication. Note: if you show this directly, great! You don't have to take the exam. Otherwise, you can show first that  and then that the identity (***) is satisfied. But you still have to take the exam in that case. This problem can be worked on jointly until Friday 10/1. Final preparation must be done individually.

manco     newton

2 Use counting methods to derive formulas for  and  . These do not have to be closed form, but they cannot be brute force. You also do not have to get them completely. I will take 'significant progress' as a project. 'Significant progress' means that you can count  in terms of  and you can count  in terms of the  's for j < r. As a carrot to get you to work on these, anyone getting these formulas will be excused from the exam. You may work jointly on these formulas until 10/1. After that, you must do the final preparation of the project on your own.

getz

3. Make up an exposition and assignment for your advanced algebra class in high school about polynomials. Pick at least 5 of the polynomial concepts we discussed and explain them in your own terms. Make up and solve at least 5 sample problems (including some word problems) involving polynomials. Make up at least 8 problems for your students to work on, including at least one 'challenge' problem. Put these at the bottom of your worksheet, which will be at least 3 pages 12 point with Maple output removed. (you can check this using the print preview under the file menu.)

overby   schroer