Please check if my method is correct. Solution : Let $$f(x) = x^8 - x^7 + x^2 -x +15 $$ Now, let $g(x)= x^8 -x^7=x^7(x - 1)$ and $h(x)= x^2 -x=x(x-1)$. Thus, $$f(x)=g(x) + h(x) + 15$$
On analyzing $g(x)$ and $h(x)$ we see that both the functions are negative between $0$ and $1$ and the maximum negative value of both the functions do not exceed $-1$. Thus, the minimum value of $f(x)$ is not less than $13$. Hence, $f(x)$ has no real root. Q.E.D