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    Roots of $z^3+2z+4$ lie outside the unit disk

    P-BGFKE

    Area
    Complex Analysis
    Topics
    Polynomials, Rouché, Zeros
    Status
    draft

    Prove that the following polynomial has its roots outside of the unit circle: \[ p(z) = z^3 + 2z + 4 .\]

    Hint: What is the maximum value of the modulus of the first two terms if \(\abs{z} \leq 1\)?

    Solution

    Big: \(M(z) = 4\) Small: \(m(z) = z^3 + 2z\). On \(\abs{z} = 1\), \[ \abs{m(z)} \leq \abs{z}^3 + 2\abs{z} = 1+2=3 < 4 = \abs{M(z)} ,\] so \(0 = Z_M = Z_{M+m} = Z_f\) in \(\DD\).

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