How is the range of a polynomial function related to the degree of the polynomial?

Answer 1

polynomials of odd degree have range (-∞,+∞)

if the polynomial has even degree, say 2k, then x^(2k) is always positive. So, depending on the leading coefficient, the range could be (-∞,N) or (N,+∞) where N will be the absolute minimum of maximum of the range.

Think of y=x^3. It extends to ∞ in both directions.

y=x^2 is a parabola, which has a vertex, and extends forever from there. This will be true of all even-degree polynomials; there will be some kind of absolute min or max, and then the sides will extend to ∞ beyond that.