  The polynomial $f_{2n}(x)=1+x+\cdots+x^{2n}$ and its minimizer on the real
line $x_{2n}=\operatorname{arg\,inf} f_{2n}(x)$ for $n\in\Bbb N$ are studied.
Results show that $x_{2n}$ exists, is unique, corresponds to $\partial_x
f_{2n}(x)=0$, and resides on the interval $[-1,-1/2]$ for all $n$. It is
further shown that $\inf f_{2n}(x)=(1+2n)/(1+2n(1-x_{2n}))$ and $\inf
f_{2n}(x)\in[1/2,3/4]$ for all $n$ with an exact solution for $x_{2n}$ given in
the form of a finite sum of hypergeometric functions of unity argument.
Perturbation theory is applied to generate rapidly converging and
asymptotically exact approximations to $x_{2n}$. Numerical studies are carried
out to show how many terms of the perturbation expansion for $x_{2n}$ are
needed to obtain suitably accurate approximations to the exact value.
