To address the question, we will first analyze the polynomial function $f(x) = x^3 - x - 6$. Here's the process:

Step 1: Find the number of real zeros

The real zeros of a polynomial function correspond to the points where the graph crosses the x-axis (i.e., the values of $x$ that make $f(x) = 0$).

We can attempt to find the zeros of the function algebraically by using either factoring or numerical methods (like the Rational Root Theorem or graphing).

Step 2: Graph the function

By graphing $f(x) = x^3 - x - 6$ using a graphing utility, we can get an approximation of the number of real zeros and their approximate locations.

Let's analyze the behavior of the function:

• As $x$ approaches large positive values ($x \to +\infty$), the cubic term $x^3$ will dominate, and the function will rise towards positive infinity.
• As $x$ approaches large negative values ($x \to -\infty$), the cubic term $x^3$ will dominate, and the function will fall towards negative infinity.

To locate the real zeros, you would graph $f(x) = x^3 - x - 6$, and look for where the curve intersects the x-axis.

Step 3: Analyze the graph options

Now let's examine the provided graph options:

• Option A: The graph has a zero at $x = -2$ and another zero at $x = 2$, as the curve passes through these points on the x-axis. It also has a maximum and minimum, with the curve descending from the left, going through a minimum in quadrant 3, rising, passing through $(0, -6)$, and continuing to fall.

• Option B: The graph passes through $(0, 0.2)$, which is not consistent with $f(0) = -6$. Therefore, this graph is not correct.

• Option C: The graph has a zero at $x = 2$, and another near $x = -2$. It also correctly passes through the point $(0, -6)$. This behavior matches the polynomial, as the function goes from negative at large $x$ values to positive at large positive $x$ values, with zeros around $-2$ and $2$.

• Option D: The curve passes through $(0, 0.2)$, which is incorrect since $f(0) = -6$. So, this graph is also incorrect.

Conclusion:

The graph that matches the description of the polynomial $f(x) = x^3 - x - 6$ is Option C.

Step 4: Number of real and imaginary zeros

• Real zeros: By analyzing the graph and solving the cubic equation $f(x) = x^3 - x - 6 = 0$, we can see that the function has three real zeros. Two of these are real roots, and one of them appears to be a multiple root near $x = -2$.

• Imaginary zeros: Since the polynomial is cubic, it must have a total of 3 roots, counting real and imaginary roots. If there are three real zeros, there are no imaginary zeros.

Final Answer:

• The function has three real zeros and no imaginary zeros.

Would you like to explore how to find the exact roots algebraically, or do you need help with any other part of the problem?

Here are five related questions for you:

1. How can you use the Rational Root Theorem to find potential rational zeros for this function?
2. What are the steps to solving cubic equations using synthetic division?
3. How do you determine the local minima and maxima of a cubic function like $f(x) = x^3 - x - 6$?
4. How can you use the first and second derivatives to analyze the behavior of the function?
5. What is the significance of the end behavior of polynomial functions?

Tip: For cubic functions, always check the signs of the coefficients and use graphing utilities to help visualize the function before attempting to solve algebraically.