A question about zeros of polynomials

[zetafunction]

POlynomials (or Taylor series ) of the form

$$P(x)= \sum_{n}a_{2n}X^{2n}$$ with $$a_{2n}\ge 0$$ strictly

have ALWAYS pure imaginary roots ??

it happens with $$sinh(x)/x$$ $$cos(x)$$ could someone provide a counterexample ? is there an hypothesis with this name ??

 

[tiny-tim]

Hi zetafunction!

Hint: start by treating it as a polynomial in x², and factor it as (x² + z₁)(x² + z₂)…(x² + z_n).

What happens if any of the zs are imaginary? ​