How to count the real roots of a quartic equation?

Question

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.

Answer 1

See http://www.jstor.org/stable/2972804?seq=1#page_scan_tab_contents

and http://mathworld.wolfram.com/DescartesSignRule.html

as well as Is there a general formula for solving 4th degree equations (quartic)?

also Quartic Equation Solution and Conditions for real roots?

Basically, the answer is no. There is not a simple and quick way to do it. There are ways, but they require a bit of work or are not guaranteed.

Answer 2

General (and simple) method for polynoms is Sturm sequence, if you know coefficients. In general case there is no methods. But if you know smth about coefficients, you can use discriminant.

Answer 3

I believe that Vieta's Formulae can help here. First, you divide the whole equation by $a$. We can redefine variables $b, c, d,$ and $e$ as $\frac{b}{a}, \frac{c}{a}, \frac{d}{a},$ and $\frac{e}{a}$ respectively. Basically, you expand the following, where $p, q, r,$ and $s$ are roots:

$$(x-p)(x-q)(x-r)(x-s)$$

And get:

$$\begin{align}b &= -(p+q+r+s) \\ c &= pq + pr + ps + qr + qs + rs \\ d &= -(pqr + pqs + prs + qrs)\\ e &= pqrs\end{align}$$

You now have $4$ variables and $4$ equations, so it is now a matter of solving this system. Admittedly, it is rather tedious, and requires quite a lot of work.