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The problem asks you to find the zeros for the polynomial function and their multiplicities. Let's go through the solution.

### Given Polynomial:
\[
f(x) = x^3 - 8x^2 + 16x
\]

### Step 1: Factor the polynomial
First, factor out the common factor \( x \):
\[
f(x) = x(x^2 - 8x + 16)
\]

Next, factor the quadratic part:
\[
x^2 - 8x + 16 = (x - 4)^2
\]

Thus, the polynomial can be rewritten as:
\[
f(x) = x(x - 4)^2
\]

### Step 2: Find the zeros
To find the zeros, set \( f(x) = 0 \):
\[
x(x - 4)^2 = 0
\]

The solutions are:
1. \( x = 0 \) (with multiplicity 1 because it's linear)
2. \( x = 4 \) (with multiplicity 2 because it's squared)

### Step 3: Behavior at each zero
- **At \( x = 0 \)**, the graph crosses the x-axis because the multiplicity is odd (1).
- **At \( x = 4 \)**, the graph touches the x-axis and turns around because the multiplicity is even (2).

### Step 4: Filling in the answers
- **There are two zeros**:
  - The multiplicity of the smallest zero (0) is **1**.
  - The multiplicity of the largest zero (4) is **2**.

- **Behavior of the function**:
  - The graph crosses the x-axis at \( x = 0 \) and touches the x-axis and turns around at \( x = 4 \).

Let me know if you'd like any further details or if you have any questions!

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Here are 5 related questions to expand on this topic:
1. What is the significance of multiplicity in determining the behavior of a graph at a zero?
2. How can we use the derivative of a polynomial to verify the nature of its turning points?
3. What is the general method for factoring higher-degree polynomials?
4. How does the Fundamental Theorem of Algebra help in finding zeros of a polynomial?
5. How does the behavior of the graph differ when a zero has an odd vs. an even multiplicity?

**Tip**: Always check the degree of your polynomial. It tells you how many roots (counting multiplicities) the function will have.

Find the zeros for the given 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) = x^3 - 8x^2 + 16x.

Learn how to find the zeros of the polynomial function f(x) = x^3 - 8x^2 + 16x and determine their multiplicities. Understand how the graph behaves at each zero by using concepts of polynomial functions and the Fundamental Theorem of Algebra. Suitable for high school students.

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