# A question about zeros of polynomials

1. Jan 5, 2010

### 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 ??

2. Jan 5, 2010

### tiny-tim

Hi zetafunction!

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

What happens if any of the zs are imaginary?