https://file.aiquickdraw.com/mathbot/file/1729468784813ktp96n6q.jpg

Let's analyze the given polynomial function:

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

### Step 1: Finding the Zeros
The zeros of the function occur when \( f(x) = 0 \). Setting the equation equal to zero:

\[
-3(x - 2)(x + 4)^2 = 0
\]

This equation will be zero when either factor equals zero:

- \( x - 2 = 0 \) gives \( x = 2 \).
- \( (x + 4)^2 = 0 \) gives \( x = -4 \).

Thus, the zeros are \( x = 2 \) and \( x = -4 \).

### Step 2: Determining Multiplicities
- \( x = 2 \) comes from the factor \( (x - 2) \), which is raised to the power of 1. Therefore, its multiplicity is **1**.
- \( x = -4 \) comes from the factor \( (x + 4)^2 \), which is raised to the power of 2. Therefore, its multiplicity is **2**.

### Step 3: Behavior at the Zeros
- At \( x = 2 \) (multiplicity 1), the graph **crosses** the x-axis because the multiplicity is odd.
- At \( x = -4 \) (multiplicity 2), the graph **touches** the x-axis and **turns around** because the multiplicity is even.

### Step 4: Answering the Multiple-Choice Question
In the multiple-choice options, the correct choice is:
- **A**: There are two zeros. The multiplicity of the largest zero (in absolute value, \( -4 \)) is **2**. The multiplicity of the smallest zero (in absolute value, \( 2 \)) is **1**.

Do you need further clarification or details?

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Here are 5 related questions for practice:
1. How would the graph change if the multiplicity of \( (x + 4)^2 \) was 3 instead of 2?
2. What are the end behaviors of the function \( f(x) = -3(x - 2)(x + 4)^2 \)?
3. How does the negative sign in front of the function \( f(x) \) affect the graph?
4. What happens to the graph if \( (x + 4)^2 \) becomes \( (x + 4)^4 \)?
5. How do you find the y-intercept of this function?

**Tip**: For polynomial functions, zeros with odd multiplicities cross the x-axis, while zeros with even multiplicities touch and turn around at the x-axis.

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.

Learn how to find the zeros of the polynomial function f(x) = -3(x - 2)(x + 4)^2, determine the multiplicity of each zero, and analyze the graph's behavior at the zeros. Perfect for students in Grades 9-12.

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