MHB Prove Root of Polynomial $P(x)=x^{13}+x^7-x-1$ Has 1 Positive Zero

[anemone]

Prove that the polynomial $P(x)=x^{13}+x^7-x-1$ has only one positive zero.

 

[kaliprasad]

Spoiler: My Solution

P(x) has one change of sign. so there is 1 positive root or number of positive roots (1 -2n) as per Descartes' rule of signs. so number of positive roots is one

 

[kaliprasad]

Spoiler: Solution 2

$P(x)=x^{13}+x^7−x−1$
$= x(x^{12}-1) +x^7-1$
$=(x-1)(x(x^{11}+ x^{10} + \cdots 1)+ (x-1)(x^6+x^5+\cdots + 1))$
$= (x-1)( x(x^{11}+ x^{10} + \cdots 1) + (x^6+x^5+\cdots + 1))$
1st term is x -1 and 2nd term is positive for positive x . so x = 1 is the only solution