Prove that all the roots of the complex polynomial \[f(z) = z^7 - 5 z^3 +12 =0\] lie between the circles \(|z|=1\) and \(|z|=2\).
Solution
On \(\abs{z} \leq 1\):
- Big: \(M(z) = 12\)
- Small: \(m(z) = z^6 - 5z^3\)
For \(\abs{z} = 1\), \[ \abs{m(z)} \da \abs{z^7-5z^3}\leq \abs{z}^7 + 5\abs{z}^3 = 6 < 12 \da \abs{M(z)} ,\] so \(0 = Z_M = Z_{f}\).
On \(\abs{z} \leq 2\),
- Big: \(M(z) = z^7\)
- Small: \(m(z) = -5z^3 + 12\)
On \(\abs{z} = 2\), \[ \abs{m(z)} \da \abs{-5z^3 + 12} \leq 5\abs{z}^2 + 12 = 32 < 128 = 2^7 \da \abs{M(z)} ,\] so \(7 = Z_M = Z_{f}\).
So \(f\) has 7 zeros in \(1 \leq \abs{z} \leq 2\).