To solve the cubic equation:

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

### Step 1: Try factoring or using the Rational Root Theorem
The Rational Root Theorem suggests testing potential rational roots, which are the divisors of the constant term (-3) divided by the divisors of the leading coefficient (-1). The possible candidates for rational roots are:

\[
\pm 1, \pm 3
\]

### Step 2: Test potential roots
Let's test \( x = 1 \) by substituting it into the equation:

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

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

### Step 3: Factor the cubic equation
Since \( x = 1 \) is a root, we can factor out \( (x - 1) \) from the cubic equation. Use synthetic division or long division to divide:

\[
-x^3 + 3x^2 + x - 3 \div (x - 1)
\]

Performing synthetic division:

\[
(-1 \quad 3 \quad 1 \quad -3) \div (1)
\]

The quotient is:

\[
-x^2 + 2x + 3
\]

Thus, the cubic equation factors as:

\[
-(x - 1)(x^2 - 2x - 3) = 0
\]

### Step 4: Solve the quadratic equation
Now, solve the quadratic \( x^2 - 2x - 3 = 0 \) using factoring:

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

### Step 5: Find the roots
The full factored form of the equation is:

\[
-(x - 1)(x - 3)(x + 1) = 0
\]

So, the solutions are:

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

### Final Answer:
The solutions to the equation are \( x = 1, x = 3, x = -1 \).

---

Would you like a detailed explanation of any of the steps, or further clarification?

Here are some related questions to explore:
1. How do you apply the Rational Root Theorem to more complex polynomials?
2. What is synthetic division, and how does it work in solving polynomials?
3. How can you solve cubic equations that don’t have rational roots?
4. What is the difference between factoring and using the quadratic formula for solving quadratics?
5. How do the discriminant and the nature of roots relate to solving quadratic equations?

**Tip:** Always check for obvious roots like \( \pm 1 \) when solving cubic or higher-degree polynomials—it can simplify the process significantly!

This page provides a detailed solution to the cubic equation -x^3 + 3x^2 + x - 3. Learn how to use the Rational Root Theorem and synthetic division to factor and solve for the roots. The problem covers key algebraic concepts, including cubic equations, factoring, and solving quadratics. Suitable for high school students in Grades 9-12.

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