Question

Give a recursive definition of the set of polynomials.

A) P0(x) = 1, Pn(x) = xⁿ
B) P0(x) = x, Pn(x) = Pn-1(x) + 1
C) P0(x) = x, Pn(x) = x * Pn-1(x)
D) P0(x) = 1, Pn(x) = Pn-1(x) * x

Answer 1

The set of polynomials can be defined recursively by using option D: P0(x) = 1 and Pn(x) = Pn-1(x) * x. This means that the 0th polynomial is a constant value of 1, and for any other polynomial, we can obtain it by multiplying the previous polynomial by the variable x.

Step-by-step explanation:

To define the set of polynomials recursively, we can use option D: P0(x) = 1 and Pn(x) = Pn-1(x) * x. This means that the 0th polynomial is a constant value of 1, and for any other polynomial, we can obtain it by multiplying the previous polynomial by the variable x.

For example, P1(x) = P0(x) * x = 1 * x = x. P2(x) = P1(x) * x = x * x = x^2. And so on.

By applying this recursive definition, we can generate a sequence of polynomials with increasing powers of x.