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The equation given is:

\[
x^3 - 6x^2 + 11x - 6 = 0
\]

To solve this cubic equation, we can use the **Rational Root Theorem** to test possible rational roots and then factor the equation if possible. 

### Step 1: Find Possible Rational Roots

The possible rational roots are factors of the constant term (−6) divided by factors of the leading coefficient (1). The factors of −6 are ±1, ±2, ±3, ±6. Therefore, the possible rational roots are:

\[
\pm 1, \pm 2, \pm 3, \pm 6
\]

### Step 2: Test Possible Roots

Let's test these possible roots by substituting them into the equation:

1. **For \( x = 1 \):**

\[
1^3 - 6(1)^2 + 11(1) - 6 = 1 - 6 + 11 - 6 = 0
\]

So, \( x = 1 \) is a root.

### Step 3: Factor the Polynomial

Since \( x = 1 \) is a root, \( x - 1 \) is a factor. We can now perform polynomial division to factor the cubic equation.

Divide \( x^3 - 6x^2 + 11x - 6 \) by \( x - 1 \):

\[
\frac{x^3 - 6x^2 + 11x - 6}{x - 1} = x^2 - 5x + 6
\]

### Step 4: Solve the Quadratic Equation

Now we solve the quadratic equation \( x^2 - 5x + 6 = 0 \) by factoring:

\[
x^2 - 5x + 6 = (x - 2)(x - 3) = 0
\]

### Step 5: Find All Roots

The roots are:
\[
x = 1, \quad x = 2, \quad x = 3
\]

### Final Answer:

The solutions to the equation \( x^3 - 6x^2 + 11x - 6 = 0 \) are:

\[
x = 1, \quad x = 2, \quad x = 3
\]

Would you like further details on any of these steps, or do you have any additional questions?

---

Here are some related questions:

1. How would you verify that \( x = 2 \) and \( x = 3 \) are also roots of the equation?
2. What is the Rational Root Theorem and how is it applied?
3. How do you perform polynomial division?
4. Can you factor a cubic equation without knowing any roots first?
5. What are the different methods for solving cubic equations?
6. How can you check if a quadratic equation is factorable?
7. What is the significance of the discriminant in quadratic equations?
8. How would you solve a cubic equation if it had no rational roots?

**Tip:** When solving cubic equations, always check for easy-to-find rational roots using the Rational Root Theorem before resorting to more complex methods.

Learn how to solve the cubic equation x^3 - 6x^2 + 11x - 6 = 0 using the Rational Root Theorem and polynomial division. This detailed guide covers finding rational roots, factorization, and solving resulting quadratic equations. Suitable for Grades 10-12.

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