Find a circle about the origin containing all the roots of the polynomial equation \(x^7 + x^2 + 1 = 0\).
Solution
Let \(z\) be a root of \[ p(z)=z^7+z^2+1. \] If \(|z|\ge2\), then \[ |z|^7=|z^2+1|\le |z|^2+1. \] But for \(r\ge2\), \[ r^7\ge 2^5r^2=32r^2>r^2+1, \] which is impossible. Therefore every root satisfies \(|z|<2\).
Thus the circle \[ \boxed{|z|=2} \] contains all roots in its interior. (A smaller optimal radius can be obtained from the positive solution of \(r^7=r^2+1\), but no optimality is requested.)