You have the equation:

\[
x^3 - 5x^2 - 3x + 1 = 0
\]

This is a cubic equation. To solve it, we can use various methods such as factoring, synthetic division, or using the cubic formula. However, for most practical purposes, numerical methods like using the Rational Root Theorem followed by synthetic division are commonly used.

### Steps to Solve:

1. **Use the Rational Root Theorem**: The possible rational roots are the factors of the constant term (1) divided by the factors of the leading coefficient (1). So the possible rational roots are: ±1.

2. **Test the possible roots**: Substitute \(x = 1\) and \(x = -1\) into the equation.

- For \(x = 1\):
  \[
  (1)^3 - 5(1)^2 - 3(1) + 1 = 1 - 5 - 3 + 1 = -6 \neq 0
  \]
  So, \(x = 1\) is **not a root**.

- For \(x = -1\):
  \[
  (-1)^3 - 5(-1)^2 - 3(-1) + 1 = -1 - 5 + 3 + 1 = -2 \neq 0
  \]
  So, \(x = -1\) is **not a root**.

Since neither \(x = 1\) nor \(x = -1\) are roots, we'll need to use numerical methods or the cubic formula to find the roots.

Would you like me to use a numerical method to find the approximate roots for this cubic equation?

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Here are 5 related questions to expand your understanding:

1. What are the methods to factorize cubic equations?
2. How is the Rational Root Theorem applied to polynomial equations?
3. What are the steps of synthetic division, and how does it work?
4. Can all cubic equations be solved exactly by factoring?
5. How does the cubic formula differ from the quadratic formula?

**Tip:** For cubic equations, if rational root testing fails, numerical methods like Newton's method can help approximate the roots efficiently.

Solve the cubic equation x^3 - 5x^2 - 3x + 1 using the Rational Root Theorem and numerical methods. This detailed guide explains key algebraic concepts, including synthetic division and the cubic formula. Ideal for high school students in Grades 11-12.

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