The expression you provided is a cubic equation:

\[
x^3 - 13x + 12 = 0
\]

Let's solve this cubic equation by factoring.

### Step 1: Check for possible rational roots using the Rational Root Theorem.

The Rational Root Theorem suggests that any rational solution to the equation should be a divisor of the constant term (12) divided by a divisor of the leading coefficient (1). This means the possible rational roots are:

\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12
\]

### Step 2: Test potential roots.

We'll test these values by substituting them into the equation:

- For \( x = 1 \):
  \[
  1^3 - 13(1) + 12 = 1 - 13 + 12 = 0
  \]
  So, \( x = 1 \) is a root.

### Step 3: Factor the cubic equation.

Since \( x = 1 \) is a root, we can factor \( x^3 - 13x + 12 \) as \( (x - 1)(x^2 + ax + b) \). To find \( a \) and \( b \), we perform polynomial division or synthetic division.

Dividing \( x^3 - 13x + 12 \) by \( x - 1 \) gives:

\[
x^3 - 13x + 12 = (x - 1)(x^2 + x - 12)
\]

### Step 4: Factor the quadratic.

Now, factor the quadratic \( x^2 + x - 12 \):

\[
x^2 + x - 12 = (x - 3)(x + 4)
\]

### Step 5: Write the complete factorization.

Thus, the full factorization of the cubic equation is:

\[
x^3 - 13x + 12 = (x - 1)(x - 3)(x + 4)
\]

### Step 6: Find the solutions.

Setting each factor equal to zero:

\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\]
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\]

### Final Answer:

The solutions to the equation \( x^3 - 13x + 12 = 0 \) are:

\[
x = 1, \, 3, \, -4
\]

Would you like more details on any of the steps, or do you have any questions?

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Here are five related questions:

1. How do you perform synthetic division to factor cubic equations?
2. Can all cubic equations be solved by factoring?
3. What are the possible rational roots for a cubic equation?
4. How do you solve a cubic equation without real rational roots?
5. How do complex roots appear in cubic equations?

**Tip:** When solving cubic equations, always check for simple rational roots first using the Rational Root Theorem, as it can simplify the process significantly.

Solve the cubic equation x^3 - 13x + 12 using the Rational Root Theorem and polynomial factorization. This detailed step-by-step guide explores how to factor cubic equations, find rational roots, and apply algebraic methods. Ideal for students in Grades 10-12.

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