I am dealing with a polynomial of degree 5, which I want to prove is positive on a certain interval.
I found a statement on Wikipedia that exactly enable me to do this. There is no reference backing that statement. If anyone knows a reference (book or article) where that statement is proven and that I can cite, I would be grateful.
The statement is made for a real polynomial of degree 4 or higher on that page of Wikipedia: ``If the discriminant is positive, the number of non-real roots is a multiple of 4''.
All I need is a reference that I can cite for that result from Wikipedia.
Denote by $\rho_{0},\dots,\rho_{\text{n}-1}$ the (a priori distinct, since we only care about the case in which the discriminant is nonvanishing) roots of the real (monic) polynomial $f$, WLOG ordered so that real roots precede nonreal roots and conjugate pairs are consecutive. Denote by $\text{disc}\left(f\right)\neq 0$ the discriminant of $f$.
By definition, $$\text{disc}\left(f\right)\ =\ \prod_{\text{i}_{0}<\text{i}_{1}} \left(\rho_{\text{i}_{1}}-\rho_{\text{i}_{0}}\right)^{2}\text{.}$$ Thus \begin{align*} \text{disc}\left(f\right)>0\ \implies\ \prod_{\text{i}_{0}<\text{i}_{1}} \left(\rho_{\text{i}_{1}}-\rho_{\text{i}_{0}}\right)\ &=\ \overline{\prod_{\text{i}_{0}<\text{i}_{1}} \left(\rho_{\text{i}_{1}}-\rho_{\text{i}_{0}}\right)}\\ &=\ \prod_{\text{i}_{0}<\text{i}_{1}} \left(\overline{\rho_{\text{i}_{1}}}-\overline{\rho_{\text{i}_{0}}}\right)\\ &=\ \left(-1\right)^{s}\prod_{\text{i}_{0}<\text{i}_{1}} \left(\rho_{\text{i}_{1}}-\rho_{\text{i}_{0}}\right) \end{align*} where $s$ denotes the number of conjugate pairs of nonreal roots. The claim immediately follows.