Geometric interpretation of the solution of the complex equation $z^4+z+2=0$

Question

Question Statement:-

If $z$ is a complex number such that $z^4+z+2=0$, show that $z$ cannot lie in the interior of the circle $|z|=1$

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Attempt at a solution :-

We are given with the equation $z^4+z+2=0$, so we can write

$$z^4+z+2=0\implies z^4+z=-2 \implies |z^4+z|=2$$

Now, from triangle inequality we get

$$|z^4+z|\le |z^4|+|z|\implies |z|^4 + |z|\ge 2$$

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My deal with the question:-

Now I did think that this inequality can hold only when $|z|\ge 1$, but I don't know why but I don't feel very comfortable with this type of proof.

So, if anyone can provide me a rigorous algebraic approach to the question. And as always more elegant solutions are welcome.

Answer 1

If $|z| \lt 1$ then $|z|^4 \le |z| \lt 1$ therefore $|z|^4 + |z| \lt 1 + 1=2$ which contradicts the inequality $|z|^4 + |z|\ge 2$