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Let $p(x)$ be a polynomial of degree $n$ with only even powers. That is, $$p(x) = \sum_{i \text{ even}}^n c_i x^i$$

I am wondering if we can say anything about the properties of the zeros of $p(x)$, other than if $z$ is a zero then $-z$ must also be a zero. (Assume in general that $c_i \in \mathbb{C}$ but if $c_i \in \mathbb{R}$ that is fine too.)

Or if $p(x)$ can be factored into some nice form.

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I think not. Indeed, pick any polynomial $q(z),$ and consider $p(z) = q(z^2).$ In effect, you are replacing each root $\rho$ of $q$ with a pair $\pm \sqrt{\rho}.$

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