Originally Posted by

**TheCoffeeMachine** I've just noticed that every polynomial of degree $\displaystyle 2k+1$, with $\displaystyle k \ne 0$, has to *at least* have one real root (because complex numbers happen in pairs).

This is true, right? It's shown in my A-level textbook that if $\displaystyle z = x+iy$ is a root of a polynomial equation with *real coefficients*, then $\displaystyle z = x-iy$ is

also a root of the polynomial equation. But I doubt showing just that would be sufficient. How can it be shown?