Show that any root $z$ of $z^4+z+3=0$ satisfies $|z|>1$ Show that any root $z$ of $z^4+z+3=0$ satisfies $|z| > 1$
My working is as such:
Let $|z| \leq 1$.
Then consider $|z^4+z| \leq |z^4| + |z| \leq 1 + 1 = 2$.
So, $|z^4 + z| \leq 2$.
Thus $-2 \leq z^4 + z \leq 2$. So $1 \leq z^4+z+3 \leq 5$.
And so $z^4+z+3 \neq 0$.
Hence if $|z| \leq 1$ there is no root $z$, and so any root $z$ must satisfy $|z| < 1$.
Is this correct?

It's better to use Rouché's theorem, with $f(z)=z^4+z$ and $g(z)=3$ then on $z=1$ we have
$$|f(z)|=|z^4+z^3|\leq2<3$$
then the number of zeros $z^4+z+3$ and $3$ are the same.
