Question

For any non-negative integer (n), let (P_n(r)) be the set of all polynomials (polynomial functions) of a real variable of degree at most (n), considered as a vector space over the field (r). Let (P(r)) be:

A. A subset of all rational numbers
B. The set of all real numbers
C. A vector space of polynomials over the real numbers of any degree
D. A group of rational polynomials

Answer 1

The term (P(r)) represents a vector space of polynomials over the real numbers with no restriction on their degree, thus the correct answer is C, a vector space of polynomials over the real numbers of any degree.

Step-by-step explanation:

For any non-negative integer (n), (P_n(r)) defines a set of polynomials with real coefficients and of degree at most (n). This set is considered a vector space over the real numbers (r), known as the field. The set (P(r)) defines the totality of these polynomials without any restriction on the degree, effectively creating a vector space of polynomials over the real numbers with potentially unlimited degree. Therefore, the correct option is C. A vector space of polynomials over the real numbers of any degree.