https://file.aiquickdraw.com/mathbot/file/1729529972775f909lusi.jpg

The polynomial function given is:

\[
f(x) = 2(x - 7)^2 (x + 4)^2 (x - 5)^3
\]

We are asked to find the zeros of the function and categorize them by their multiplicity.

### Step-by-step solution:

1. **Identify the factors of the polynomial:**
   - The factor \( (x - 7)^2 \) gives a zero at \( x = 7 \) with multiplicity 2.
   - The factor \( (x + 4)^2 \) gives a zero at \( x = -4 \) with multiplicity 2.
   - The factor \( (x - 5)^3 \) gives a zero at \( x = 5 \) with multiplicity 3.

2. **List the zeros based on their multiplicity:**
   - **Multiplicity one**: There are no zeros with multiplicity 1, so the answer here is **None**.
   - **Multiplicity two**: The zeros with multiplicity 2 are \( 7 \) and \( -4 \).
   - **Multiplicity three**: The zero with multiplicity 3 is \( 5 \).

### Final answers:
- **Zero(s) of multiplicity one**: None.
- **Zero(s) of multiplicity two**: \( 7, -4 \).
- **Zero(s) of multiplicity three**: \( 5 \).

Would you like more details or have any questions?

Here are 5 related questions to expand on this:

1. How do multiplicities affect the shape of a polynomial's graph?
2. Can a polynomial have zeros with multiplicity higher than three?
3. What is the significance of the constant factor in a polynomial (like 2 in this case)?
4. How would the graph behave at a zero with an even multiplicity versus an odd multiplicity?
5. Can a polynomial have negative multiplicity for any of its factors?

**Tip**: A zero of even multiplicity causes the graph to "bounce off" the x-axis at that point, while an odd multiplicity allows it to cross the axis.

Suppose that the polynomial function f is defined as follows: f(x) = 2(x - 7)^2(x + 4)^2(x - 5)^3. List each zero of f according to its multiplicity in the categories below.

Learn how to find the zeros and their multiplicities for the polynomial function f(x) = 2(x - 7)^2(x + 4)^2(x - 5)^3. Understand how to categorize zeros based on their multiplicity with a step-by-step solution. This problem involves key polynomial concepts suitable for high school students.

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