Now consider polynomials of the form ax b where ab n how

Question

Question: 1. In this problem we will count polynomials.

(a) Consider polynomials of the form ax, where a ∈ N. How many such polynomials are there?

(b) Now consider polynomials of the form ax + b, where a,b ∈ N. How many such polynomials are there?

(c) Continue considering, and this time examine polynomials of the form ax²+bx+c, where a,b,c ∈ N. How many such polynomials are there?

(d) Finally, consider polynomials of the form anxn + an-1x^n-1 + ··· + a₁x+a₀. How many such polynomials are there?

2. Challenge: Is the set of polynomials with integer coefficients countable or uncountable?