Why does the polynomial equation $1 + x + x^2 + \cdots + x^n = S$ have at most two solutions in $x$? Américo Tavares pointed out in his answer to this question that finding the ratio of a geometric progression only from knowledge of the sum of its first $n+1$ terms $S = 1+x+x^2+\cdots+x^n$ amounts to solving a polynomial of degree $n$. This suggested to me that there might be up to $n$ real solutions of $x$ for a given sum, but I could not find any. In fact, it turned out that the following fact is true:
For $n \ge 1$ and $S \in \mathbb{R}$, the polynomial equation $x^n + x^{n-1} + \cdots + x + 1 = S$ has at most two real solutions.
A corollary is that if $n$ is odd, there is exactly one real solution. I was only able to prove this using a rather contrived geometric argument based on the shape of the graph of $y = x^{n+1}$. Is there a simple, direct (and ideally, intuitive) proof of this fact?

The roots are also roots of
$x^{n+1} - Sx + S - 1 = 0$ 
which we get by multiplying your equation by $x-1$.
This polynomial ($x^{n+1} - Sx + S-1$), as we move from $x = -\infty$ to $x = \infty$ is either 


*

*Monotonically increasing, and thus has at most one real root.

*Monotonically decreasing, and then monotonically increasing and hence can have at most two real roots.

*Monotonically increasing, then decreasing and then again increasing (happens only when $n$ is even). In which case there are at most three real roots, one of which is $1$. So for $S \ne n+1$, the original equation does not have more than two solutions. If $S=n+1$ and $n$ is even, then the turning points are $-1$ and $1$ and the value of the polynomial at $-1$ is positive. So the only roots are $1$ and a root which is $< -1$.
This can be seen by looking at its derivative, which is an increasing function for odd $n$, and for even $n$, it is positive, then possibly negative (depending on $S$) and then positive again, as we move from $x = -\infty$ to $x = \infty$.
