To solve the equation x^3 - 2x + 1 = 0, we can use various methods such as factoring, graphing, or using numerical methods. However, in this case, we will use the Rational Root Theorem and synthetic division to find the rational roots of the equation.

The Rational Root Theorem states that if a polynomial equation has a rational root, it will be of the form p/q, where p is a factor of the constant term (in this case, 1) and q is a factor of the leading coefficient (in this case, 1).

The factors of 1 are ±1, and the factors of 1 are ±1. Therefore, the possible rational roots are ±1.

We can now use synthetic division to test these possible roots:

For x = 1:
1 | 1 -2 1
| 1 -1
| 1 -1 0

Since the remainder is 0, x = 1 is a root of the equation.

Now, we can divide the original equation by (x - 1) to obtain a quadratic equation:

(x - 1)(x^2 - x - 1) = 0

Setting each factor equal to zero, we have:

x - 1 = 0 or x^2 - x - 1 = 0

Solving x - 1 = 0, we find x = 1.

To solve x^2 - x - 1 = 0, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = -1, and c = -1. Plugging these values into the quadratic formula, we get:

x = (-(-1) ± √((-1)^2 - 4(1)(-1))) / (2(1))
= (1 ± √(1 + 4)) / 2
= (1 ± √5) / 2

Therefore, the solutions to the equation x^3 - 2x + 1 = 0 are x = 1, x = (1 + √5) / 2, and x = (1 - √5) / 2.