A question about zeros of polynomials

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Polynomials of the form P(x) = ∑_{n} a_{2n} x^{2n} with a_{2n} ≥ 0 strictly have pure imaginary roots. This conclusion is supported by the analysis of functions such as sinh(x)/x and cos(x). The discussion suggests treating the polynomial as a function of x² and factoring it into (x² + z1)(x² + z2)…(x² + zn) to explore the implications of imaginary roots.

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POlynomials (or Taylor series ) of the form

[tex]P(x)= \sum_{n}a_{2n}X^{2n}[/tex] with [tex]a_{2n}\ge 0[/tex] strictly

have ALWAYS pure imaginary roots ??

it happens with [tex]sinh(x)/x[/tex] [tex]cos(x)[/tex] could someone provide a counterexample ? is there an hypothesis with this name ??
 
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Hi zetafunction! :smile:

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? :wink:​
 

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