https://file.aiquickdraw.com/mathbot/file/17299112620420koojm5p.jpg

The polynomial function given is:

\[
f(x) = 7x(x + 6)(x - 3)^2(x - 9)
\]

To find the zeros of \( f(x) \) and their multiplicities, let's analyze each factor:

1. **Zero from \( x \):**
   - \( x = 0 \)
   - Multiplicity: 1 (since \( x \) appears as a single factor)

2. **Zero from \( x + 6 \):**
   - \( x = -6 \)
   - Multiplicity: 1 (since \( x + 6 \) appears as a single factor)

3. **Zero from \( (x - 3)^2 \):**
   - \( x = 3 \)
   - Multiplicity: 2 (since \( (x - 3) \) is squared)

4. **Zero from \( x - 9 \):**
   - \( x = 9 \)
   - Multiplicity: 1 (since \( x - 9 \) appears as a single factor)

### Summary of Zeros and Their Multiplicities:

- **Multiplicity one:** \( 0, -6, 9 \)
- **Multiplicity two:** \( 3 \)
- **Multiplicity three:** None

So, fill in the answers as follows:
- **Zero(s) of multiplicity one:** 0, -6, 9
- **Zero(s) of multiplicity two:** 3
- **Zero(s) of multiplicity three:** None

Would you like more details on the solution, or do you have any questions?

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### Related Questions
1. How do you determine the multiplicity of a zero in a polynomial?
2. What effect does a zero with even multiplicity have on the graph of a polynomial?
3. How can we factor a polynomial to determine its zeros?
4. What is the significance of each factor's exponent in a polynomial function?
5. How does changing the multiplicity of a zero affect the shape of the graph near that zero?

### Tip
When analyzing a polynomial, focus on the powers of each factor to quickly identify the multiplicities of its zeros.

Suppose that the polynomial function f is defined as follows: f(x) = 7x(x + 6)(x - 3)^2(x - 9). List each zero of f according to its multiplicity in the categories below.

Analyze the polynomial function f(x) = 7x(x + 6)(x - 3)^2(x - 9) to identify zeros and their respective multiplicities. Understand concepts related to polynomials and multiplicities. Suitable for high school students in Grades 10-12.

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