How the determine the number of real positive roots of a polynomial?

The Problem

For a polynomial (of degree 3 or 4) with real coefficients how can we determine the number of real positive roots?

or phrased a different way

What conditions enure that a polynomial (of degree 3 or 4) with real coefficients has 0,1,2,3,or 4 positive real roots?

Context

I have a set of polynomials which have complicated coefficients which consists of several constants. I can determine the conditions when these coefficients are positive or negative. I am interested in only the positive real roots of these polynomials.

Possible Theorems to Use

Use a combination of Descartes' rule of signs, and Rolle's theorem?

Notes

• These posts with similar titles Number of real positive roots of a polynomial? Number of real positive roots of a polynomial? as the polynomials in these questions have numbers as their coefficients (rather than unknown constants) or have a specific structure.


• I am not interested in finding the explicit roots to these polynomials. I can solve them using Mathematica and other computer algebra systems. Determining when the roots are positive and real is the real problem.


• I am not sure what tags to use for this post, suggestions and edits are welcome.

Answer

What you want is
Sturm's theorem:

https://en.wikipedia.org/wiki/Sturm%27s_theorem

This gives a method
of computing
the number of real roots
of a polynomial
in any interval
of the real line.