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Let $z$ be a complex number such that $|z|=1$. Find the minimum of $\left| z^4+z+\frac{1}{2} \right|$

I squared the modulus and used the polar form of $z$ so that $\overline{z}$ would vanish. Then I had to find the minimum of $\cos a+2\cos 3a+\cos 4a$, which I expressed as a polinomial in $\cos a=x: 8x^4+8x^3-8x^2-5x+1$. I tried using derivatives but calculations got really messy...

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    Take $z=\cos\theta+i\sin\theta$. – 2017-01-22
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    or you can set $$z=x+iy$$ – 2017-01-22
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    `calculations got really messy` The *derivative* of that polynomial has a rational root, which leaves a quadratic to solve for the other two. Not too pretty, indeed, yet solvable. – 2017-01-22

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