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The given polynomial function is:

\[
f(x) = 6x(x + 9)(x - 5)^2(x + 12)^3
\]

### Step 1: Finding the Zeros of the Function
The zeros of a polynomial are found by setting each factor to zero and solving for \(x\).

- From the factor \(6x\), we get:
  \[
  x = 0
  \]
  (Multiplicity = 1, since \(x\) appears as \(x^1\)).

- From the factor \((x + 9)\), we get:
  \[
  x = -9
  \]
  (Multiplicity = 1, since \((x + 9)\) appears as \((x + 9)^1\)).

- From the factor \((x - 5)^2\), we get:
  \[
  x = 5
  \]
  (Multiplicity = 2, since \((x - 5)\) is squared).

- From the factor \((x + 12)^3\), we get:
  \[
  x = -12
  \]
  (Multiplicity = 3, since \((x + 12)\) is cubed).

### Step 2: Listing Zeros by Multiplicity

- **Zero(s) of multiplicity one**: \(x = 0\), \(x = -9\)
- **Zero(s) of multiplicity two**: \(x = 5\)
- **Zero(s) of multiplicity three**: \(x = -12\)

### Final Answer:
- Zero(s) of multiplicity one: \(0, -9\)
- Zero(s) of multiplicity two: \(5\)
- Zero(s) of multiplicity three: \(-12\)

Would you like more details on any part of this solution? Here are some related questions to deepen understanding:

1. What does the multiplicity of a zero mean geometrically on a graph?
2. How do you factor more complex polynomials with higher degrees?
3. What happens to the shape of the polynomial function around zeros with different multiplicities?
4. How do changes in multiplicity affect the behavior of the function near its zeros?
5. How can you use synthetic division to confirm the zeros of a polynomial?

**Tip**: Zeros with even multiplicities "touch" the x-axis but do not cross it, while zeros with odd multiplicities cross the x-axis.

Suppose that the polynomial function f(x) = 6x(x + 9)(x - 5)^2(x + 12)^3. List each zero of f according to its multiplicity in the categories provided.

Learn how to determine the zeros of the polynomial function f(x) = 6x(x + 9)(x - 5)^2(x + 12)^3 and list them based on multiplicity. Explore key polynomial concepts such as roots and their multiplicities, which affect the graph's behavior. Suitable for high school students studying algebra.

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