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The given polynomial function is:

\[
f(x) = -8(x - 5)^3 (x + 1)^4 x^2
\]

### Finding the Zeros and their Multiplicities:
1. **Step 1: Identify the factors.**
   The function is factored as:
   \[
   f(x) = -8(x - 5)^3 (x + 1)^4 x^2
   \]
   The factors of \( f(x) \) are \( (x - 5) \), \( (x + 1) \), and \( x \).

2. **Step 2: Find the zeros.**
   To find the zeros, set each factor equal to 0:
   - \( x - 5 = 0 \)  →  \( x = 5 \)
   - \( x + 1 = 0 \)  →  \( x = -1 \)
   - \( x = 0 \)

3. **Step 3: Determine the multiplicity of each zero.**
   The multiplicity of a zero is the exponent on its corresponding factor:
   - The factor \( (x - 5)^3 \) gives the zero \( x = 5 \) with multiplicity 3.
   - The factor \( (x + 1)^4 \) gives the zero \( x = -1 \) with multiplicity 4.
   - The factor \( x^2 \) gives the zero \( x = 0 \) with multiplicity 2.

### Final Answer:
- Zero at \( x = 5 \), multiplicity 3.
- Zero at \( x = -1 \), multiplicity 4.
- Zero at \( x = 0 \), multiplicity 2.

Would you like more details on the concept of multiplicity or have any other questions?

Here are 5 related questions for further exploration:
1. What happens to the graph at a zero with even multiplicity versus odd multiplicity?
2. How do you determine the end behavior of a polynomial based on its degree and leading coefficient?
3. What effect does the multiplicity of a zero have on the shape of the graph at that point?
4. Can a polynomial function have a zero with multiplicity 1?
5. How would the function change if the leading coefficient \( -8 \) were positive?

**Tip:** A zero with odd multiplicity causes the graph to cross the x-axis, while a zero with even multiplicity causes the graph to touch but not cross the x-axis.

Find the zeros of the polynomial function and state the multiplicity of each.

Learn how to find the zeros and their multiplicities of the polynomial function f(x) = -8(x - 5)^3 (x + 1)^4 x^2. Understand the concept of multiplicity and the factor theorem in polynomial functions. Suitable for Grades 10-12.

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