To solve the cubic equation \( X^3 - 4X^2 + 5X - 2 = 0 \), we look for the roots of the equation. There are a few ways to approach this, including:

1. **Trial and error for rational roots**: Using the Rational Root Theorem.
2. **Numerical methods**: Such as using Newton's method or directly solving using software.

Let's first test possible rational roots and then proceed with other methods if necessary.

### 1. Rational Root Theorem:
The Rational Root Theorem suggests that any rational root of the polynomial equation will be a factor of the constant term (here, -2) divided by a factor of the leading coefficient (here, 1). The possible rational roots could be \( \pm 1, \pm 2 \).

I'll check these possible roots.The roots of the cubic equation \( X^3 - 4X^2 + 5X - 2 = 0 \) are:

\[
X = 1, \quad X = 2
\]

These are the only real roots of the equation. Since the equation is cubic and has only these two real roots, one of them is a repeated root.

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How can we factor a cubic polynomial after finding its roots?
2. What are the implications of having repeated roots in a polynomial equation?
3. How do the Rational Root Theorem and the remainder theorem relate?
4. Can you find complex roots of cubic equations, and how?
5. What methods are there for solving higher-degree polynomials beyond cubic?

**Tip:** When a cubic equation has two distinct roots, one of the roots is typically repeated or there's a pair of complex conjugates.

Learn how to solve the cubic equation X³ - 4X² + 5X - 2 step-by-step. This problem involves polynomial equations and the application of the Rational Root Theorem to find roots. Suitable for advanced high school students (Grades 10-12).

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