Find the circle about the origin containing all roots of $X^7+X^2+1=0$ We need to  find the circle about the origin containing  all roots of $X^7+X^2+1=0$,
I am not getting any hint where to start from. It is a problem of Munkres General topology section $56$, chapter Fundamental Group.
Since the question is tagged "complex analysis" I thing a good approach could be to use Rouche's Theorem, so

Hints: 

(1) Put $\,f(z):=z^2+1\;,\;\;g(z):=z^7\,$ 

(2) Check that on $\,C:=\left\{z\in\Bbb C\;;\;|z|=\frac{5}{4}\,\right\}\,$ , for example,  we have $\,|f(z)|&lt;|g(z)|\,$

(3) How many roots does $\,g(z)\,$ have within $\,C\,$ ?...
Hint: From the equation, you know that $X$ must be negative and less than negative one. You need to determine the relationship between $X^7$ and $X^2$ that allows their summation ($X^7$ will be negative) to be equal to $-1$.