I have a polynomial of degree six: $x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0=0$ for which I know it will always have only two real roots and 4 complex. The coefficients $a_5\ldots{a}_0$ will change, creating a family of this kind of polynomials. I can find their roots numerically (using Matlab).
Is there a way to evaluate the distance between the real roots (in an analytical form) ? I want to prove that this distance cannot be higher than a threshold. Thank you.