Based on some experiments, I find that the following two statements are correct. But I can not prove this. At the same time, I still can not find the counterexmaples.


Let $p(x)=x^{n}+a\_{2}x^{n-2}+a\_{3}x^{n-3}+\dots+a\_{n-1}x+a\_{n}$ be a polynomial with interger coefficients, where $a\_{k}\geq0$ for every even $k$ and $a\_{k}\leq0$ otherwise. Note here that the coefficient of $x^{n-1}$ is equal to 0. Suppose that there exists some odd number $p$ such that $a\_{p}<0$ and $a\_{p-1}>0$. Then the following two statements should be true:


1. $p(x)$ has at most one nonzero real root.
2. $p(x)$ has no pure imaginary zeros, i.e. $p(x)$ has no zero in the form $\alpha\textrm{i}$, where $\alpha\neq0$ and $\textrm{i}^{2}=-1$.


Thanks for your time.


I am sorry for losing a condition that $a\_{2}\geq5$.



Two problems about the zeros of a prescribed polynomial
$(x^3-1)(x^3-2)(x^2+a^2)=x^8+a^2x^6-3x^5-3a^2x^3+2x^2+2a^2$ has both two different non-zero real roots and $\pm ai$ as roots.


**EDIT** polynomial adjusted for adjusted conditions. If $a\_2$ needs to be at least $b$, set $a$ so $a^2\geq b$. :)


