2021/068) Prove that the polynomial equation $x^8-x^7+x^2-x+15=0$ has no real 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