To solve the equation x^3 - 3x^2 + 3x + 1 = 0, you can use the Rational Root Theorem to find possible rational roots. The Rational Root Theorem states that if a polynomial equation has a rational root p/q, where p is a factor of the constant term (1 in this case) and q is a factor of the leading coefficient (1 in this case), then p/q is a possible rational root.

In this case, the possible rational roots are ±1. We can test these values by substituting them into the equation:

For x = 1:
(1)^3 - 3(1)^2 + 3(1) + 1 = 1 - 3 + 3 + 1 = 2 ≠ 0

For x = -1:
(-1)^3 - 3(-1)^2 + 3(-1) + 1 = -1 - 3 - 3 + 1 = -6 ≠ 0

Since neither x = 1 nor x = -1 satisfy the equation, there are no rational roots for this equation. Therefore, the equation x^3 - 3x^2 + 3x + 1 = 0 does not have any rational solutions.