Problem of the Week: A Couple of Squares

Editor’s Note: The Department of Mathematics at Gettysburg College hosts a problem of the week challenge to determine each semester’s Paul Mugabi problem-solving award recipient(s).  Each week’s entries are scored by a faculty judge, and winner(s) from each week will receive a Problem Of the Week (P.O.W.) button.  The Gettysburgian is not involved in or responsible for accepting or evaluating students’ submissions to this contest.

Problem of the week

THE RULES:

The contest is open to all Gettysburg College students. Up to three people may work together on a submission. Make sure your name is on your submission and that any sources are properly cited. Send solutions to bkennedy@gettysburg.edu.  This problem was posted on Friday, September 17 and solutions are due on Friday, September 24 by 5:00 p.m.

THE PROBLEM: 

Professor Kennedy asked Professor Conceição for help calculating the square of a particular nonzero polynomial p(x). They decided to each do the calculation separately and then compare their answers.

Professor Kennedy (interested in dynamics) and Professor Conceição (interested in algebra) did not discuss the problem carefully enough ahead of time, however: Professor Conceição calculated the “square of p(x)” by calculating p(x) · p(x) (the polynomial p(x) multiplied by itself), and Professor Kennedy calculated the “square of p(x)” by calculating p(p(x)) (the polynomial p(x) composed with itself).

Nevertheless, when the two professors compared their answers, they each had exactly the same polynomial! Assuming that neither Professor Conceição nor Professor Kennedy made a mistake, what can you conclude about the polynomial p(x)?

Author: Gettysburgian Staff

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