# [SOLVED]Nature of roots challenge

#### anemone

##### MHB POTW Director
Staff member
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 $x > 1$
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 $x < 1$