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Mentallic

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This is a polynomial of infinite degree, and the theorem that says a polynomial must have at least one root in the complex field, and thus this extends to a polynomial of nth degree having n roots (not necessarily distinct).

However, for [itex]e^x=0[/itex] there are zero roots. Why is this possible when, clearly by the power series for [itex]e^x[/itex] it should be an infinite degree polynomial with infinite roots (in the complex plane)?