We are tasked with factoring the cubic polynomial:

\[
-2x^3 + 15x^2 - 31x + 12
\]

### Step 1: Factor out the common factor
Since the leading coefficient is negative, we factor out \(-1\) to simplify the process:

\[
-1(2x^3 - 15x^2 + 31x - 12)
\]

### Step 2: Use the Rational Root Theorem
We use the Rational Root Theorem to test for possible rational roots. The possible rational roots are factors of the constant term (12) divided by factors of the leading coefficient (2). The possible rational roots are:

\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm \frac{1}{2}
\]

### Step 3: Test for roots using synthetic division
We begin testing the possible rational roots. Let's start with \(x = 3\):

\[
\text{Perform synthetic division of } 2x^3 - 15x^2 + 31x - 12 \text{ by } x - 3.
\]

Performing synthetic division:

\[
\begin{array}{r|rrrr}
3 & 2 & -15 & 31 & -12 \\
  &   & 6  & -27 & 12 \\
\hline
  & 2 & -9  & 4  & 0  \\
\end{array}
\]

Since the remainder is 0, \(x = 3\) is a root, and the quotient is \(2x^2 - 9x + 4\).

### Step 4: Factor the quadratic \(2x^2 - 9x + 4\)
Now we factor \(2x^2 - 9x + 4\). We look for two numbers that multiply to \(2 \times 4 = 8\) and add to \(-9\). These numbers are \(-8\) and \(-1\), so we factor the quadratic as follows:

\[
2x^2 - 9x + 4 = (2x - 1)(x - 4)
\]

### Step 5: Write the complete factorization
Now, we combine all the factors:

\[
-2x^3 + 15x^2 - 31x + 12 = -(x - 3)(2x - 1)(x - 4)
\]

Thus, the factorization is:

\[
-(x - 3)(2x - 1)(x - 4)
\]

Let me know if you need further clarification or additional details!

### Follow-up Questions:
1. How does the Rational Root Theorem help in factoring polynomials?
2. What is synthetic division, and how is it used in polynomial factorization?
3. Why did we factor out \(-1\) at the beginning of the process?
4. Can you factor a higher-degree polynomial using similar techniques?
5. What are some other methods for solving cubic polynomials?

**Tip:** Always check your result by multiplying the factors back together to ensure correctness!

Learn how to factor the cubic polynomial −2x^3 + 15x^2 − 31x + 12 step-by-step using the Rational Root Theorem and synthetic division. This detailed guide covers polynomial factorization, quadratic equations, and essential algebraic techniques. Ideal for high school students in Grades 9-12.

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