Complex Analysis – How Many Roots Have Modulus Less Than 1?

Question

If roots of the equation
$$x^7 – 4x^3 + x + 1=0$$

are plotted on the Argand plane, how many of them have distance from the origin less than $1$?

I found, by plotting the rough curve of $y= x^7 – 4x^3 + x + 1$ that it has three real roots, out of which two of them have modulus greater than $1$ and one has modulus less than $1$. But I don't know how to do the same for non – real roots. The answer given in my book is that $3$ roots have modulus less than $1$.

Answer 1

Rouches theorem: let $f,g$ be holomorphic in open set $U$ and $C$ boundary in $U$, of a disc inside $U$. If $|f|>|g|$ for all $z$ on the circle $C$ then $f$ and $f+g$ have same number of zeros inside $C$ (counted with multiplicity).

Let $g(z)=z^7+z+1$, and $f(z)=-4z^3$. Both are clearly holomorphic functions.

Then for $|z|=1$ i.e. $z$ on the unit circle (since it is in your question), we have $$|f(z)|=|-4\cdot 1|=4 \mbox{ and } |g(z)|\leq |z|^7+|z|+1=3.$$ Thus, $|g(z)|<|f(z)|$ for all $z$ inside unit circle $C$.

Can you complete the solution now?