Let's solve the equation \(x^3 - 4x + 1 = 0\) to find its roots.

This is a cubic equation, and solving it analytically involves either factoring or using methods like the Rational Root Theorem or Cardano's method. However, in most cases, numerical methods or approximations are used to find the roots.

### Steps:

1. **Check for Rational Roots:**
   The Rational Root Theorem suggests that any rational root, if it exists, must be a divisor of the constant term (1) divided by the leading coefficient (1). The possible rational roots are \( \pm 1 \).

2. **Test Potential Rational Roots:**
   - For \( x = 1 \): 
     \[
     1^3 - 4(1) + 1 = 1 - 4 + 1 = -2 \quad \text{(Not a root)}
     \]
   - For \( x = -1 \):
     \[
     (-1)^3 - 4(-1) + 1 = -1 + 4 + 1 = 4 \quad \text{(Not a root)}
     \]

Since neither \( x = 1 \) nor \( x = -1 \) is a root, there are no simple rational roots.

3. **Numerical Approximation:**
   To find the roots, we use numerical methods such as Newton's method, or use a calculator or software to approximate them.

   The approximate roots are:
   - \( x_1 \approx -1.5321 \)
   - \( x_2 \approx 0.55496 \)
   - \( x_3 \approx 0.97713 \)

### Final Answer:
The roots of the equation \(x^3 - 4x + 1 = 0\) are approximately \(x \approx -1.5321\), \(x \approx 0.55496\), and \(x \approx 0.97713\).

Would you like a detailed explanation of how to use Newton's method to find these roots, or do you have any other questions?

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**Relative Questions:**

1. What is Newton's method for finding roots of an equation?
2. How can the Rational Root Theorem help in solving polynomial equations?
3. What are the properties of cubic functions, and how do they affect the number of roots?
4. How does the behavior of a cubic function differ from that of a quadratic function?
5. What are the conditions for a cubic equation to have all real roots?

**Tip:** When solving cubic equations, always check for possible rational roots first, as they can simplify the process significantly.

Learn how to solve the cubic equation x^3 - 4x + 1 = 0 with a detailed step-by-step solution. This problem covers cubic equations, root finding using numerical methods, and applications of the Rational Root Theorem. Suitable for students in Grades 10-12.

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