An exercise from my Complex Analysis course (mostly focused on integration):
Show that all the roots of $f(z)=z^7-5z^3+12$ lie inside the annulus $1<|z|<2$.
Knowing that, being a polynomial function, $f(z)$ has 7 roots and no poles in $\mathbb{C}$, I thought I could try using the argument principle and show that $$\frac1{2\pi i}\oint\limits_{|z|=2} \frac{f'(z)}{f(z)}dz = 7$$ and $$\frac1{2\pi i}\oint\limits_{|z|=1} \frac{f'(z)}{f(z)}dz = 0$$ but it doesn't really seem to be a clever approach to me, since the integration would likely require finding the roots of $f(z)$ first... Any ideas?