To solve the cubic equation \( x^3 - 2x + 1 = 0 \), we can use several methods. Since this is a cubic equation, finding an exact solution typically involves either factoring, using the Rational Root Theorem, or applying numerical methods.

### Step 1: Check for Rational Roots
By the Rational Root Theorem, possible rational roots are factors of the constant term (1) divided by factors of the leading coefficient (1). The possible rational roots are \( \pm 1 \).

- Testing \( x = 1 \):
  \[
  1^3 - 2(1) + 1 = 1 - 2 + 1 = 0
  \]
  So, \( x = 1 \) is a root.

### Step 2: Factor the Polynomial
Since \( x = 1 \) is a root, we can factor the cubic equation as:
\[
x^3 - 2x + 1 = (x - 1)(ax^2 + bx + c)
\]
To find \( a \), \( b \), and \( c \), perform polynomial division of \( x^3 - 2x + 1 \) by \( x - 1 \).

Let's do that division:

\[
\frac{x^3 - 2x + 1}{x - 1}
\]

This will give us a quadratic factor, which we can then solve for the remaining roots.

### Step 3: Solve the Remaining Quadratic Equation
After performing the polynomial division, solve the resulting quadratic equation.

I'll work through these steps now.The division yields:

\[
x^3 - 2x + 1 = (x - 1)(x^2 + x - 1)
\]

Now, the equation becomes:

\[
(x - 1)(x^2 + x - 1) = 0
\]

We already know that \( x = 1 \) is one root. Now, we need to solve the quadratic equation \( x^2 + x - 1 = 0 \).

### Step 4: Solve the Quadratic Equation
We can solve the quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For \( x^2 + x - 1 = 0 \), the coefficients are \( a = 1 \), \( b = 1 \), and \( c = -1 \). Let's calculate the roots.The roots of the quadratic equation \( x^2 + x - 1 = 0 \) are:

\[
x = \frac{-1 + \sqrt{5}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{5}}{2}
\]

### Final Solution:
The roots of the original cubic equation \( x^3 - 2x + 1 = 0 \) are:

1. \( x = 1 \)
2. \( x = \frac{-1 + \sqrt{5}}{2} \) (approximately 0.618)
3. \( x = \frac{-1 - \sqrt{5}}{2} \) (approximately -1.618)

These are the exact roots of the equation.

Would you like more details on any step, or have any other questions?

### Related Questions:
1. How do you apply the Rational Root Theorem to other cubic equations?
2. What is the geometric meaning of the roots of a cubic function?
3. Can cubic equations always be factored like this?
4. How can you use numerical methods to approximate roots?
5. What are the applications of cubic equations in real-world problems?

**Tip:** Knowing how to factor a polynomial can significantly simplify solving higher-order equations!

Learn how to solve the cubic equation x^3 - 2x + 1 = 0 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the Rational Root Theorem and polynomial division. Suitable for advanced high school students and college-level algebra courses.

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