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 $-1$.