Why can't you have more turning points than the degree? I get that each degree can correspond to a factor.   $x^5=(x+a)(x+b)(x+c)(x+d)(x+e)$ and that results in 4 turning points, so the graph can "turn around" and hit the next zero.    Why can't a curve have more turning points than zeros?   
In the graph below, there are 4 roots, so degree is 4, but way more turning points than 4.    What gives?   Are those additional turning points represented by imaginary roots?


A quintic cannot have more than $4$ turning points, for the reason K. Jiang explains. On the other hand, a quintic (or even a cubic) can have more turning points than zeroes. For example, $x(x-1)(x-2)(x-3)(x-4)+2$ has four turning points, but only three zeroes.

