The most direct method that I can currently think of is the following:
Differentiate the polynomial, and make a substitution $w=x^2$. This gives us $$7w^3-50w^2+15,$$ a cubic equation in $w$. There is a general formula for solving for the zeroes of a cubic equations, hence we can find the exact roots of this equation, and in turn, we can find the exact roots of $f'(x)$, which tells us where local max/min occur. Checking the values of the function at these points will tell us how many zeroes the polynomial has.
This is a little bit of a long method and probably not the cleanest; I'll see if I can think of a simpler method, but this is certainly one approach to the problem.