# I have given the following nonconstant complex polynomial h ( x ) = x n </ms

I have given the following nonconstant complex polynomial $h\left(x\right)={x}^{n}+{a}_{n-1}{x}^{n-1}+\dots +{a}_{1}x+{a}_{0}$. In the lecture our Prof. told us that using the minimal principle one could find $z\in \mathbb{C}$ such that $h\left(z\right)=0$
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Suppose $h\left(z\right)$ is a polynomial of degree at least 1 and $h\left(z\right)$ has no roots.
Consider $g\left(z\right)=\frac{1}{h\left(z\right)}.$ If h has no roots, g is analytic.
$|g\left(0\right)|>0$ (or else we have found a root and have a contradiction.)
Over the domain $|z|\le R$ the maximum modulus theorem says the maximum of $|g\left(z\right)|$ must lie on the circle $|z|=R$
But if as |z| gets to be large, $|h\left(z\right)|$ goes to infinity.
Hence for large enough R, $|z|=R$ implies $0<|g\left(z\right)|<|g\left(0\right)|$ which is a contradiction.