Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 2 Exercises 2.8
Our main tool is Corollary 2.3.
Solution: By Corollary 2.3, if $x^8-4x^5+13x^3-7x+1=0$ has a rational solution, then this rational solution must be a divisor of $1$ which means it can only be $\pm 1$.
If $x=1$, we have
$$
x^8-4x^5+13x^3-7x+1=1-4+13-7+1=4\ne 0.
$$ Hence $1$ is not a solution.
If $x=1$, we have
$$
x^8-4x^5+13x^3-7x+1=1+4-13+7+1=0.
$$ Hence $-1$ is a solution.
The only rational solution of $x^8-4x^5+13x^3-7x+1=0$ is $-1$.