Question

Consider polynomials of degree at most n, each of whose coefficients is an element of Z. How many such polynomials satisfy:

a) 2^n
b) n^2
c) 3^n
d) 2^n+1

Answer 1

The number of distinct polynomials of a certain degree with integer coefficients is determined by the range of integer values that can be assigned to each coefficient, which is derived from the given terms.

Step-by-step explanation:

The question deals with the number of distinct polynomials of degree at most n with coefficients in the set of integers (Z). When the coefficients are integers, we can derive possible values for coefficients of the polynomial terms from the powers of the given numbers. For instance, in the case where the term 2^n is given, we can infer that the coefficients may be any of the integers in the set {0, 1, ..., 2^n-1}. Similarly, the other cases follow with each coefficient having n^2, 3^n, and 2^(n+1) distinct possibilities, respectively.