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The given polynomial is \( P(x) = x^3 - x - 6 \), and the task is to find all the zeros (roots) of the polynomial.

### Step 1: Apply the Rational Root Theorem
To find the possible rational roots, the Rational Root Theorem suggests that we check factors of the constant term (-6) over the leading coefficient (1). This gives the following possible rational roots:
\[
\pm 1, \pm 2, \pm 3, \pm 6
\]

### Step 2: Test Possible Rational Roots
We can test these values by substituting them into the polynomial.

Let's start with \( x = 2 \):
\[
P(2) = 2^3 - 2 - 6 = 8 - 2 - 6 = 0
\]
Thus, \( x = 2 \) is a root.

### Step 3: Perform Polynomial Division
Now that we know \( x = 2 \) is a root, we can factor out \( (x - 2) \) from the polynomial. We divide \( x^3 - x - 6 \) by \( x - 2 \) using synthetic division or long division.

Using synthetic division:

\[
\begin{array}{r|rrrr}
2 & 1 & 0 & -1 & -6 \\
  &   & 2 & 4  & 6 \\
\hline
  & 1 & 2 & 3  & 0 \\
\end{array}
\]
The quotient is \( x^2 + 2x + 3 \), so we can rewrite the polynomial as:
\[
P(x) = (x - 2)(x^2 + 2x + 3)
\]

### Step 4: Solve the Quadratic
Now, solve the quadratic \( x^2 + 2x + 3 = 0 \) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1 \), \( b = 2 \), and \( c = 3 \), so:
\[
x = \frac{-2 \pm \sqrt{2^2 - 4(1)(3)}}{2(1)} = \frac{-2 \pm \sqrt{4 - 12}}{2} = \frac{-2 \pm \sqrt{-8}}{2} = \frac{-2 \pm 2i\sqrt{2}}{2}
\]
\[
x = -1 \pm i\sqrt{2}
\]

### Final Answer
The zeros of the polynomial are:
\[
x = 2, \, x = -1 + i\sqrt{2}, \, x = -1 - i\sqrt{2}
\]

Would you like more details on any part of this solution? Here are five related questions:

1. How do you apply synthetic division in solving polynomials?
2. Can you explain the Rational Root Theorem in more depth?
3. Why does the quadratic formula work for finding complex roots?
4. What are imaginary numbers, and why do they appear in this solution?
5. How do we verify the roots of a polynomial?

**Tip:** When solving polynomials, always test rational roots first to simplify the equation before moving on to more advanced methods like quadratic formulas or factoring.

Find all zeros of the polynomial: P(x) = x^3 - x - 6. Enter your answers as a comma-separated list.

Learn how to find all zeros of the polynomial P(x) = x^3 - x - 6 using the Rational Root Theorem, synthetic division, and the quadratic formula. This detailed guide covers solving for real and complex roots, making it ideal for high school students in Grades 9-12.

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