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The given polynomial function is:

\[
f(x) = x^3 - 6x^2 + 9x
\]

### Step 1: Factor the polynomial
Factor the expression to find the zeros:

\[
f(x) = x(x^2 - 6x + 9)
\]

Now, factor the quadratic part \(x^2 - 6x + 9\):

\[
f(x) = x(x - 3)(x - 3)
\]

This can also be written as:

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

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

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

Thus, the solutions are:
- \(x = 0\)
- \(x - 3 = 0 \Rightarrow x = 3\)

### Step 3: Multiplicity of the zeros
- \(x = 0\) has a multiplicity of 1 (because \(x\) appears only once).
- \(x = 3\) has a multiplicity of 2 (because \((x - 3)^2\) means \(x - 3\) is squared).

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

### Final Answer:
The zeros are:
- \(x = 0\) (multiplicity 1, crosses the x-axis)
- \(x = 3\) (multiplicity 2, touches the x-axis and turns around)

Do you need any additional explanations or details?

Here are five related questions to deepen understanding:
1. How does the multiplicity of a zero affect the graph at that point?
2. Can a cubic polynomial have more than three distinct zeros?
3. What is the relationship between the degree of a polynomial and the number of zeros it has?
4. How can you determine the end behavior of the graph of a cubic function?
5. How do you factor higher-degree polynomials?

**Tip:** Always check the multiplicity of each zero to understand the behavior of the graph near the x-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 for f(x) = x^3 - 6x^2 + 9x.

Learn how to find the zeros and multiplicity of the cubic polynomial f(x) = x^3 - 6x^2 + 9x. Includes step-by-step factoring, analysis of graph behavior at each zero, and discussion of multiplicity. Suitable for high school students.

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