https://file.aiquickdraw.com/mathbot/file/1727757479675ognouvup.jpg

The given polynomial function is:

\[
f(x) = x^3 - 8x^2 + 16x
\]

### Step 1: Factor the polynomial
We start by factoring the function:

\[
f(x) = x(x^2 - 8x + 16)
\]

Next, factor the quadratic part:

\[
x^2 - 8x + 16 = (x - 4)^2
\]

So, the fully factored form of the polynomial is:

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

### Step 2: Find the zeros
To find the zeros, set \( f(x) = 0 \):

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

Thus, the zeros are \( x = 0 \) and \( x = 4 \).

### Step 3: Determine the multiplicities
- The zero \( x = 0 \) comes from the factor \( x \), which has a multiplicity of 1.
- The zero \( x = 4 \) comes from the factor \( (x - 4)^2 \), which has a multiplicity of 2.

### Step 4: Analyze the graph
- Since the zero at \( x = 0 \) has a multiplicity of 1, the graph will **cross** the x-axis at this point.
- Since the zero at \( x = 4 \) has a multiplicity of 2, the graph will **touch** the x-axis and turn around at this point.

### Answer to the question:
- The zeros are \( 0 \) and \( 4 \).
- The multiplicity of the smallest zero (\( x = 0 \)) is 1.
- The multiplicity of the largest zero (\( x = 4 \)) is 2.

Thus, the correct answer is **B**.

Let me know if you need further details or explanations! 

Here are five related questions:
1. What is the relationship between a zero's multiplicity and the graph's behavior at that zero?
2. How does the factorization of polynomials help in solving higher-degree equations?
3. What happens when a polynomial has a zero with an odd multiplicity?
4. How can synthetic division help in finding the factors of a polynomial?
5. Can a polynomial have complex zeros if all its coefficients are real?

**Tip:** Always verify the behavior of a polynomial graph by analyzing the multiplicities of its zeros; higher even multiplicities cause the graph to bounce at the axis!

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 f(x) = x^3 - 8x^2 + 16x and determine their multiplicity. This detailed solution includes factoring, analyzing zeros, and understanding how the graph behaves at each zero.

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