The given equation is:

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

To solve this cubic equation, we can use synthetic division or the Rational Root Theorem to find possible rational roots. Let's begin by testing possible values for \(x\) that could simplify this equation.

### Step 1: Test for possible rational roots
The Rational Root Theorem tells us that possible rational roots are the factors of the constant term (−3) divided by the factors of the leading coefficient (1). So, possible rational roots are:

\[
\pm 1, \pm 3
\]

### Step 2: Test \(x = 1\)

Substitute \(x = 1\) into the equation:

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

Thus, \(x = 1\) is a root of the equation.

### Step 3: Factor the cubic equation
Now that we know \(x = 1\) is a root, we can factor \(x - 1\) out of the cubic polynomial using synthetic division:

Performing synthetic division of \(x^3 + 3x^2 - x - 3\) by \(x - 1\) gives:

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

### Step 4: Solve the quadratic equation
Next, we solve the quadratic \(x^2 + 4x + 3 = 0\). Factoring this quadratic:

\[
x^2 + 4x + 3 = (x + 1)(x + 3)
\]

### Step 5: Find all roots
Now, we have the factored form of the cubic equation:

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

Thus, the solutions are:

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

### Final Answer:
The roots of the equation \(x^3 + 3x^2 - x - 3 = 0\) are:

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

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Do you need more details on any step, or have any further questions?

Here are 5 related questions for practice:
1. Solve the equation \(x^3 - 6x^2 + 11x - 6 = 0\).
2. Factor the cubic equation \(x^3 + 2x^2 - 5x - 6 = 0\).
3. Solve \(x^3 - 4x^2 + x + 6 = 0\).
4. Find the roots of \(x^3 + x^2 - 2x - 2 = 0\).
5. Solve \(x^3 + x^2 - 4x - 4 = 0\).

**Tip:** When solving cubic equations, the Rational Root Theorem is a good starting point for finding possible rational solutions.

Explore the solution of the cubic equation x^3 + 3x^2 − x − 3 = 0 using the Rational Root Theorem. Learn step-by-step how to find the roots by factoring and solving quadratic equations. Suitable for high school students in Grades 10-12.

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