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The polynomial function given is:

\[
f(x) = -9(x+5)(x-8)^2
\]

### Step 1: Finding the Zeros
To find the zeros, set \( f(x) = 0 \):

\[
-9(x+5)(x-8)^2 = 0
\]

This equation equals zero when either factor equals zero:
1. \( x+5 = 0 \Rightarrow x = -5 \)
2. \( (x-8)^2 = 0 \Rightarrow x = 8 \)

Thus, the zeros are \( x = -5 \) and \( x = 8 \).

### Step 2: Multiplicity of Each Zero
- The zero \( x = -5 \) comes from the factor \( (x + 5) \), which is raised to the first power. Therefore, its **multiplicity is 1**.
- The zero \( x = 8 \) comes from the factor \( (x - 8)^2 \), which is squared. Therefore, its **multiplicity is 2**.

### Step 3: Behavior of the Graph at Each Zero
- **At \( x = -5 \)**: Since the multiplicity is 1 (odd), the graph **crosses** the x-axis at this point.
- **At \( x = 8 \)**: Since the multiplicity is 2 (even), the graph **touches** the x-axis and **turns around** at this point.

### Conclusion:
- The zeros are \( x = -5 \) and \( x = 8 \).
- Multiplicity of \( x = -5 \) is 1, and the graph crosses the x-axis here.
- Multiplicity of \( x = 8 \) is 2, and the graph touches the x-axis and turns around here.

Would you like further details on any part of this?

Here are 5 related questions to expand on this:
1. How does the leading coefficient of a polynomial affect the graph's end behavior?
2. Can a polynomial have a zero with multiplicity greater than 2?
3. What is the significance of the degree of a polynomial in relation to its zeros?
4. How can you factor higher-degree polynomials to find their zeros?
5. What happens to the graph of a polynomial if the multiplicity of a zero is 3?

**Tip**: The behavior at zeros can often be predicted by their multiplicity: odd multiplicities lead to crossings, while even multiplicities result in touching and turning.

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) = -9(x+5)(x-8)^2

This webpage covers how to find the zeros of a polynomial function, determine the multiplicity of each zero, and understand the graph's behavior at each zero. The function in this example is f(x) = -9(x+5)(x-8)^2, and the solution explains how the graph crosses or touches the x-axis based on the multiplicity of each zero. Suitable for high school students (Grades 9-11).

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