Question 1 - Roots

Consider a polynomial of fifth degree p(x) = a5 x5 + a4 x4 + a3 x3 + a2 x2 + a1 x + a0. Then, consider the polynomial q(x) related to p(x) in the following way:

q(x) = a0 x5 + a1 x4 + a2 x3 + a3 x2 + a4 x + a5

a) Create two tables, one for polynomial p(x) and one for q(x), where one element of the table is the list of roots, in decimal form, of p(x) and q(x), respectively (for given values of the coefficients). The coefficients a0, a1, a2, a3, a4, a5 must take all possible values from -5 to 5 with a step of 2 (that is, each coefficient will parse -5,-3,-1,1,3 and 5). Make sure the elements in the two tables are paired up (that is, for example the tenth element in the table for p(x) corresponds with the 10th element in the table for q(x) where q(x) is created from p(x) as above mentioned).

b) Randomly create a list of 25 indexes (that is a list of random integers that refer to positions in the tables). Create two new tables (lists) with the corresponding elements from the two tables. Create a Manipulate object that displays the points from each of the two tables for the selected index (that is, the parameter for Manipulate command is the index in the newly created lists). Make sure to use to different colors for the elements from the two newly created sublists.

Make sure your horizontal axis is fixed. Use PlotRange[] for this purpose.

c) For the tables created in a), verify if it is true that roots of corresponding polynomials are reciprocal of each other.

Interested? Related to this is the story on how Euler found the sum of the series: 1 + 1/4 + 1/ 9 + 1 /16 + 1/ 25 + ... Just google: “How Euler found the sum of the reciprocals of squares”