Can we find sufficient conditions in which this equation have only three distinct real roots

Question

Let us consider the polynomial equation:

$$ξ₁x⁸+ξ₂x⁷+ξ₃ x⁶+ξ₄ x⁵+ξ₅ x⁴+ξ₆ x³+ξ₇ x²+(ξ₈-1) x+ξ₉ =0$$

where $ξ_{i}$ are real coefficients.

My question is: Can we find sufficient conditions in which this equation have only three distinct real roots.

Answer 1

Hint:

Since a polynomial of the form:

$p(z)=\sum_{k=0}^m a_{k}z^{k}$

Can be expressed as the following where $\beta_{1},...,\beta_{m}$ are $m$ roots of the polynomial:

$p(z)=a_{0}(z-\beta_{1})...(z-\beta_{m})$

Assume $5$ of the $8$ roots are complex and expand the polynomial. See if there exist conditions in which there could be 3 real and 5 complex roots.

I.e.

$\xi_{1}(x-a)(x-b)(x-c)(x-(\alpha_{1}+i\beta_{1})(x-(\alpha_{2}+i\beta_{2}))...(x-(\alpha_{5}+i\beta_{5}))$

Expand this out and see if there exist conditions in which this could be a polynomial with real coefficients.