TITLE: How can I guarantee the unique positive root of this polynomial?
QUESTION [9 upvotes]: How can I guarantee the unique positive root of this polynomial?
I have two polynomial, 
$$
x^{n+1} + x^n - 1 =0 
$$
and 
$$
x^{n+1} - x^n - 1 =0 
$$
respectively, where $n\in\mathbb{N}$. I have tried for the cases from $n=1$ to $100$. For every calculation, I have found unique positive root for each polynomial with the help of MATHEMATICA, but I couldn't prove it mathematically. How can I prove these polynımials have unique positive root ?

REPLY [2 votes]: From $x^n = \frac{1}{( x \pm 1 )}$
explore the interception of two functions:
     $x^n$ (always positive growing for $x>0$ ) whatever $n>0$
and  $\frac{1}{x+1}$ (always decreasing from $1$ to $0$ for $x>0$ )
  or $\frac{1}{x-1}$ (always decreasing from $\infty$ to $0$ for $x>1$  ) 
the interception belongs always and necessarily to the first quadrant 
( explore the graphs in wolframalpha.com , varying the $n$ )
solve $x^n = \frac{1}{( x + 1 )}$  and $n=2$
solve $x^n = \frac{1}{( x - 1 )}$  and $n=2$