6 1. Fundamentals- Show that, for any positive integer k,
(1+ t)(1 + t2)(1 + t4>... (1+ t2y = 1 + t + t2 + t3 +. * * + t?- Characterize those polynomials p(t) for which
(4 p(t) = d-4
(b) p(t) = --d-t).- As seen in Question 4, it is not always the case that p o q = q o p for
polynomials p and q. If p o q = q o p, then q is said to commute with
p under composition. Determine all polynomials p(t) which commute
under composition with t2, i.e. for which p(t2) = [p(t)12.
Explorations
E.l. Square of a Polynomial. The square of a polynomial is the product
of a polynomial with itself. Normally, the square has more nonzero terms
than the polynomial itself. Show that this always occurs for polynomials
of degrees 1, 2 and 3 having more than one term. Find a polynomial with
more than one term whose square has exactly the same number of terms as
the polynomial. Is it possible to find a polynomial whose square actually
has fewer terms?
E.2. (a) Let p(t) = at2 + bt + c be any quadratic polynomial. Verify that~(1) + ~(4) + ~(6) + ~(7) = ~(2) + ~(3) + ~(5) + ~(8).
(b) Partition the set of numbers { 1,2,3,... ,14,15,16} into two sets such
that, given any cubic polynomial p(t) with integer coefficients, the sum of
the numbers p(k) where k ranges over one of the two sets is the same as
the sum where k ranges over the other.
(c) Let m be a positive integer. It is a remarkable fact that the numbers
from 1 to 2”‘+l inclusive can be subdivided into two subsets A and B
such that, for any polynomial p(t) of degree not exceeding m, the sum
of the values of the polynomials over the numbers in A is equal to the
sum of the values over the numbers in B. Show that we can reduce the
problem to finding sets A and B for which the sum of the kth powers of
the numbers in one set is equal to the sum of the kth powers for the other,
for k = 0, 1,2,... , m.
(d) This situation can be generalized. If d and m are any integers with
d 2 2, the set of numbers from 1 to d”‘+’ can be subdivided into d disjoint
subsets such that, for any polynomial of degree not exceeding m, the sum
of its values over any of the subsets is the same.
(e) Problem (a) can be generalized in another way. Consider the question
of looking for disjoint sets (al, ~2,... , a,) and (bl, b2,... , b,,) of integers for