To solve the cubic equation \( x^3 - 3x + 2 = 0 \), we can attempt to factor the equation by finding rational roots. We can use the **Rational Root Theorem**, which suggests that any possible rational root is a factor of the constant term divided by a factor of the leading coefficient.

Given the equation \( x^3 - 3x + 2 = 0 \):

1. **Identify coefficients**:
   - The constant term is \( 2 \).
   - The leading coefficient is \( 1 \).

2. **Possible rational roots**:
   The possible rational roots are \( \pm 1 \), \( \pm 2 \) (the factors of 2).

3. **Test these values** by substituting them into the equation to see if they satisfy it:
   - For \( x = 1 \):
     \[
     1^3 - 3(1) + 2 = 1 - 3 + 2 = 0
     \]
     So, \( x = 1 \) is a root.

4. **Factor the polynomial**:
   Since \( x = 1 \) is a root, we can factor \( x^3 - 3x + 2 \) as \( (x - 1) \) times the quotient of the division of \( x^3 - 3x + 2 \) by \( x - 1 \).

   Performing the division:

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

5. **Factor the quadratic polynomial** \( x^2 + x - 2 \):
   \[
   x^2 + x - 2 = (x - 1)(x + 2)
   \]

6. **Final factorization**:
   The full factorization is:
   \[
   x^3 - 3x + 2 = (x - 1)(x - 1)(x + 2)
   \]

7. **Roots**:
   Set each factor equal to zero:
   \[
   x - 1 = 0 \quad \text{(repeated root)}
   \]
   \[
   x + 2 = 0
   \]

   So, the roots are \( x = 1 \) (repeated root) and \( x = -2 \).

### Summary of the roots:
- \( x = 1 \) (with multiplicity 2)
- \( x = -2 \)

Would you like to see the detailed steps of the factorization process or any further analysis of these roots?

Here are some related questions you might consider:

1. What does it mean for a root to have multiplicity greater than 1?
2. How do the roots of the equation \( x^3 - 3x + 2 = 0 \) relate to the graph of the function?
3. How would you solve a cubic equation that doesn't factor nicely?
4. What is the significance of the derivative at the repeated root?
5. How can the factorization of a cubic polynomial help in graphing it?

**Tip:** When dealing with polynomials, always check for the multiplicity of roots, as it affects the behavior of the graph at those points.

Learn how to solve the cubic equation x^3 - 3x + 2 = 0 with a detailed step-by-step solution. This problem covers cubic equations, factorization using the Rational Root Theorem, and finding roots of polynomials. Suitable for students in Grades 10-12.

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