Show that all roots of $z^4+z^3+1$ of of modulus less than $3/2$ by using Rouche's Theorem. I was wondering if you could help me understand how to to apply Rouche's Theorem to this Complex Analysis problem:

Show that all roots of $z^4+z^3+1$ of of modulus less than $3/2$ by using Rouche's Theorem.
There are a few ways to state Rouche's theorem, but here's one that will work in this case: if $f$ and $g$ are holomorphic on a domain containing a circle $C$ and its interior, and if $|f|&gt;|g|$ on $C$, then $f$ and $f+g$ have the same number of zeros inside $C$.

In this case, take $f(z)=z^4$, $g(z)=z^3+1$, and $C=\{|z|=\frac{3}{2}\}$.
Consider the polynomial $P(z)=z^4+z^3+1$ and the curve $\gamma$ that traces out the circle $|z|&lt;3/2$. Define $f(z):=z^4$ and $g(z):=z^3+1$. Then $P(z)=(f+g)(z)$. Since $f$ and $g$ are also polynomials, they are analytic on and inside $\gamma$. Note also that $|f(z)|&gt;|g(z)|$ for all $z\in \gamma$, because:
        \begin{align*}
   |f(z)|&amp;=|z^4|\leq (3/2)^4=81/16\\
   |g(z)|&amp;=|z^3+1|\leq |z^3|+1\leq (3/2)^3+1=70/16
  \end{align*}
        Along the boundary $\gamma$, these upper bounds are maximized. So, along $\gamma$, we have that $|f(z)|&gt;|g(z)|$. 
        Recall by Rouche's Theorem that if $f$ and $g$ are analytic inside and on a regular closed curve $\gamma$ and that $|f(z)|&gt;|g(z)|$ for all $z\in \gamma$, then the number of zeros, including order, of $f+g$ inside $\gamma$ is equal to the number of zeros, including order, of $f$ inside gamma.
        Since $f(z)=z^4$, there exists a zero of order $4$ at $z=0$ inside $\gamma$. Hence, $P(z)=(f+g)(z)$ also has $4$ zeros inside of $\gamma$ as well. Since these four zeros are inside the circle $|z|&lt;3/2$, then the zeros are of modulus less than $3/2$. Further, by the Fundamental Theorem of Algebra, we have that since the degree of $P(z)$ equals four, then $P(z)$ has exactly four zeros on in its domain. However, using Rouche's Theorem, we have shown that all four of the zeros of $P(z)$ are of modulus less than $3/2$, as desired.