Algebra and Number Theory

Tuesday, July 17

"It's in the cards: Predicting the coefficients of a polynomial of nth degree"

Ryota presented a set of data and we calculated the first and second differences. We found that the second differences were constant. Our work has shown that if the first difference is linear then the second difference is constant. Doing back substitution (up and over), we were able to see Pascal’s triangle pattern emerge. We thought this was pretty neat, but still would like to find a generalized formula for polynomials of a higher degree, namely the nth degree. We worked with linear, quadratic and cubic and saw some simple general forms, but when we worked on 4th and 5th degree polynomials things began to not be so simple. We did make some progress, but discussed if the cost of solving it in this fashion versus the worth of its effect. At the end of our session we briefly discussed examples of problems that we could use to illustrate our findings. Bob who joined us today from California provided a very interesting problem which involves the number of points on a circle and the number of regions formed by the connection of these points. At first it appears the relationship is 2^x, but by observing the data we found this to be false. On Thursday, we plan to continue working on trying to fit a polynomial for the data generated from Bob’s problem.

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This material is based upon work supported by the National Science Foundation under Grant No. 0314808 and Grant No. ESI-0554309. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.