Suppose I have a quartic equation with real coefficients, such as:
$$a x^4 +b x^3+c x^2+d x +e=0$$
I want to know the number of its real roots. Search engines lead me to symbolic expressions for all the roots, and these can be produced by CAS packages like Mathematica, but these results are too long and complex (in both senses) to be of use to me.
I would hope there is a compact/efficient method to count the real roots of a real quartic equation, similar to the way the discriminant of a quadratic polynomial tells us the number of real roots of a real quadratic equation.