Here are a few other questions for you to discuss with your classmates.
(a) How many local extrema could a polynomial of degree n have? How few local extrema can it have?
(b) Could a polynomial have two local maxima but no local minima?
(c) If a polynomial has two local maxima and two local minima, can it be of odd degree? Can it be of even degree?
(d) Can a polynomial have local extrema without having any real zeros?
(e) Why must every polynomial of odd degree have at least one real zero?
(f) Can a polynomial have two distinct real zeros and no local extrema?
(g) Can an x-intercept yield a local extrema? Can it yield an absolute extrema?
(h) If the y-intercept yields an absolute minimum, what can we say about the degree of the polynomial and the sign of the leading coefficient?