In Chapter 1 of Polynomials by Victor Prasolov, Springer, 2001, the following theorem is proved. (p.3)
Theorem 1.1.4 (Ostrovsky). Let $f(x)=x^{n}-b_{1}x^{n-1}-\cdots -b_{n}$, where all the numbers $b_{i}$ are non-negative and at least one of them is nonzero. If the greatest common divisor of the indices of the positive coefficients $b_{i}$ is equal to $1$, then $f$ has a unique positive root $p$ and the absolute value of the other roots are $<$ p.
The following is one of the Problems to Chapter 1 (p.41).
Problem 1.5 - Find the number of real roots of the following polynomials
a) ...
b) $nx^{n}-x^{n-1}-\cdots -1$
Question: How to solve this Problem?
Added: $nx^{n}-x^{n-1}-\cdots -1=0$ $\Leftrightarrow x^{n}-\dfrac{1}{n}x^{n-1}-\cdots -\dfrac{1}{n}=0$
Added 2: Sturm's Theorem.