This is a polynomial of degree 6 with leading coefficient -3.
Descartes' rules of signs tells you that the maximum # of real zeros is 2.
In fact, the most useful way to answer this question when graphing is so easy is to graph the equation which shows immediately that there are, in fact, no real zeros!
Without using the "black box" as my friend and I call the computer or graphing calculator, there are a couple of things you can find out pretty quickly. This polynomial is dominated by the 6th power term as |x| becomes large, which tells you that the value becomes negatively unbounded quickly as x increases in either direction. Secondly both the 1st and 2nd derivative are 0 at x = 0; this means there is a point of inflection with an horizontal tangent at x = 0 Lastly looking at the first derivative tells you that the maximum is close to one where the polynomial is negative, i.e. it never crosses the x-axis!
