MHB Can the polynomial equation $x^8-x^7+x^2-x+15=0$ have real roots?

[anemone]

Prove that the polynomial equation $x^8-x^7+x^2-x+15=0$ has no real solution.

 

[kaliprasad]

Spoiler: My solution

We have $x^8-x^7 + x^2 -x + 15 = x^7(x-1) + x(x-1) + 15$

each term is positive for $x > 1$ so LHS is greater than 0 so no solution for $ x > 1$

for x = 1 LHS = 15 so x = 1 is not a solution

Further $x^8-x^7 + x^2 -x + 15 = (15- x) + x^2(1-x^5) + x^8 $

Each term is positive for $x < 1$ so LHS is greater than 0 so no solution for $ x < 1$

Hence no real solution