Find the zeros for the polynomial function and give the multiplicity for each zero.  State whether the graph crosses the​ x-axis or touches the​ x-axis and turns around at each zero.
​f(x)equals x cubed plus 6 x squared minus 4 x minus 24=x3+6x2−4x−24

To find the zeros of the polynomial function \( f(x) = x^3 + 6x^2 - 4x - 24 \), we'll follow these steps:

1. **Factor the polynomial** to find its zeros.
2. **Determine the multiplicity** of each zero.
3. **Analyze the behavior** of the graph at each zero (whether it crosses the x-axis or touches it and turns around).

### Step 1: Factor the Polynomial
We first attempt to factor the cubic polynomial. 

Since this is a cubic polynomial, we can start by checking if there is an obvious root (zero) by using the Rational Root Theorem, which suggests that any rational root must be a factor of the constant term (\(-24\)) divided by the leading coefficient (\(1\)). The possible rational roots are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24 \).

We can test these by substituting them into the polynomial.

Let's test \( x = 2 \):

\[
f(2) = (2)^3 + 6(2)^2 - 4(2) - 24 = 8 + 24 - 8 - 24 = 0
\]

So, \( x = 2 \) is a root.

Now, we can factor \( x - 2 \) out of \( f(x) \):

Using synthetic division:

\[
\begin{array}{r|rrrr}
2 & 1 & 6 & -4 & -24 \\
  &   & 2 & 16 & 24 \\
\hline
  & 1 & 8 & 12 & 0 \\
\end{array}
\]

This gives us:

\[
f(x) = (x - 2)(x^2 + 8x + 12)
\]

### Step 2: Factor the Quadratic Polynomial
Next, factor the quadratic \( x^2 + 8x + 12 \) if possible:

The quadratic formula can be used here:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where \( a = 1 \), \( b = 8 \), and \( c = 12 \):

\[
x = \frac{-8 \pm \sqrt{64 - 48}}{2} = \frac{-8 \pm \sqrt{16}}{2} = \frac{-8 \pm 4}{2}
\]

So:

\[
x = \frac{-8 + 4}{2} = -2 \quad \text{or} \quad x = \frac{-8 - 4}{2} = -6
\]

Thus, the quadratic factors as:

\[
x^2 + 8x + 12 = (x + 2)(x + 6)
\]

### Step 3: Write the Fully Factored Form
The fully factored form of \( f(x) \) is:

\[
f(x) = (x - 2)(x + 2)(x + 6)
\]

### Step 4: Determine Zeros and Their Multiplicity
The zeros of the function are:
- \( x = 2 \) with multiplicity 1.
- \( x = -2 \) with multiplicity 1.
- \( x = -6 \) with multiplicity 1.

### Step 5: Analyze the Behavior at Each Zero
- **At \( x = 2 \)**: The zero has an odd multiplicity (1), so the graph crosses the x-axis at \( x = 2 \).
- **At \( x = -2 \)**: The zero has an odd multiplicity (1), so the graph crosses the x-axis at \( x = -2 \).
- **At \( x = -6 \)**: The zero has an odd multiplicity (1), so the graph crosses the x-axis at \( x = -6 \).

### Final Answer:
- The zeros are \( x = 2 \), \( x = -2 \), and \( x = -6 \), each with a multiplicity of 1.
- The graph crosses the x-axis at each zero.

Would you like further details or have any other questions?

Here are five related questions you might find interesting:
1. How do you use the Rational Root Theorem to find potential zeros?
2. What is the significance of multiplicity in determining the behavior of a graph?
3. Can a polynomial of degree 3 have fewer than three real zeros?
4. How does the shape of the graph of a cubic function generally look like?
5. How does synthetic division work, and when is it used?

**Tip:** Always check for common factors first when factoring polynomials, as it can simplify the process significantly.

Discover how to find the zeros and their multiplicity for the polynomial function f(x) = x^3 + 6x^2 - 4x - 24. This detailed solution covers factoring, determining multiplicity, and analyzing graph behavior at each zero. Suitable for Grades 11-12.

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