https://file.aiquickdraw.com/mathbot/file/1728995372563u3gz6ipf.jpg

The polynomial function given in the image is:

\[
f(x) = (x - 4)^2 (x - 8)^3 (x + 5)^3
\]

We need to find the zeros of this polynomial and their multiplicities.

### Step-by-Step Solution:

1. **Identify the factors**: The function is already factored into terms \((x - 4)^2\), \((x - 8)^3\), and \((x + 5)^3\).

2. **Zeros of the polynomial**: The zeros are the values of \(x\) that make each factor zero.

   - From \((x - 4)^2\), the zero is \(x = 4\).
   - From \((x - 8)^3\), the zero is \(x = 8\).
   - From \((x + 5)^3\), the zero is \(x = -5\).

3. **Multiplicities**:
   - The factor \((x - 4)^2\) has a multiplicity of 2.
   - The factor \((x - 8)^3\) has a multiplicity of 3.
   - The factor \((x + 5)^3\) has a multiplicity of 3.

### Answer:
- **Zero(s) of multiplicity one**: None.
- **Zero(s) of multiplicity two**: \(4\).
- **Zero(s) of multiplicity three**: \(8, -5\).

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

### Related Questions:
1. What happens to the graph of a polynomial at zeros with different multiplicities?
2. How would you factor a polynomial to find its zeros and multiplicities?
3. What does it mean for a zero to have a multiplicity of 1, 2, or 3?
4. How does multiplicity affect the behavior of a polynomial at the corresponding zero?
5. Can a polynomial have a zero with a multiplicity higher than 3?

### Tip:
A zero with an odd multiplicity (like 3) causes the graph to cross the x-axis, while a zero with an even multiplicity (like 2) makes the graph touch the x-axis and turn around.

Suppose that the polynomial function f is defined as follows:

f(x) = (x - 4)^2 (x - 8)^3 (x + 5)^3

List each zero of f according to its multiplicity in the categories below.
If there is more than one answer for a multiplicity, separate them with commas. If there is no answer, click on "None."

Learn how to find the zeros and their multiplicities for the polynomial function f(x) = (x - 4)^2 (x - 8)^3 (x + 5)^3. This problem covers key algebraic concepts including polynomial functions, zeros, and multiplicities. Suitable for students in Grades 10-12.

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