Solve below equestion by factoring cubic equestion
 
x3 – 6x2 + 11x - 6

We are tasked with factoring the cubic equation:

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

### Step 1: Try possible rational roots using the Rational Root Theorem.
The Rational Root Theorem suggests testing factors of the constant term (−6) over the leading coefficient (which is 1). The possible rational roots are:
\[
\pm 1, \pm 2, \pm 3, \pm 6
\]

### Step 2: Test these possible roots by substitution.
Let's test \(x = 1\):

\[
1^3 - 6(1^2) + 11(1) - 6 = 1 - 6 + 11 - 6 = 0
\]
Thus, \(x = 1\) is a root.

### Step 3: Perform synthetic division to factor out \((x - 1)\).

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

\[
\begin{array}{r|rrrr}
 1 & 1 & -6 & 11 & -6 \\
   &   & 1  & -5 & 6 \\
\hline
   & 1 & -5 & 6  & 0 \\
\end{array}
\]

The quotient is \(x^2 - 5x + 6\).

### Step 4: Factor the quadratic \(x^2 - 5x + 6\).
Now, factor \(x^2 - 5x + 6\):

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

### Step 5: Combine all factors.
Thus, the complete factorization of the cubic polynomial is:

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

### Final Solution:
The equation factors as:
\[
(x - 1)(x - 2)(x - 3) = 0
\]

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

Would you like more details or have any questions about the process?

Here are 5 related questions to expand on this:

1. How can you verify that all roots are correct for this cubic equation?
2. What is the Rational Root Theorem and how does it apply in other problems?
3. How do you perform synthetic division step-by-step?
4. What are the common patterns to recognize when factoring quadratics?
5. How would you solve a cubic equation that has no rational roots?

**Tip**: Always test possible rational roots before using more complex methods to factor cubic equations.

Solve the equation by factoring the cubic equation: x^3 – 6x^2 + 11x - 6

Learn how to solve the cubic equation x^3 – 6x^2 + 11x - 6 by factoring. This solution walks through using the Rational Root Theorem, synthetic division, and factoring quadratic equations. Suitable for students in Grades 10-12.

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