Zeros of particular polynomials

Question

Let's have the equation $M^{2n}+DM^nK^n+K^{2n}=L^2$ all non-zero natural numbers and n≥2.Let's also have the polynomial $p(x)=x^{2n-1}+ax^{2n-2}+a^2x^{2n-3}+.....+a^{n-1}x^n+b^nx^{n-1}+ab^nx^{n-2}+a^2b^nx^{n-3}+...+a^{n-1}b^n$ where $a=M$ and $b^n=K^{2n}(a^n+2)+2K^n(a^n+1)+a^n$. Does anyone know for which values of n the polynomials p(x) have all their zeros complex numbers?

Answer 1

Your polynomial has degree $2n-1$, which is odd. Therefore it has at least one real root, no matter what $n$ and your other parameters are.