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The problem is asking you to find the zeros of the function \( z(x) = -4x^2 (x - 1)^3 (x + 2)^2 \) and state their multiplicities.

### Steps to find the zeros:

1. **Factor the function:**
   The function is already factored:
   \[
   z(x) = -4x^2 (x - 1)^3 (x + 2)^2
   \]

2. **Find the zeros:**
   The zeros occur when any of the factors are equal to zero:
   - \( x^2 = 0 \) gives \( x = 0 \).
   - \( (x - 1)^3 = 0 \) gives \( x = 1 \).
   - \( (x + 2)^2 = 0 \) gives \( x = -2 \).

3. **State the multiplicities:**
   The multiplicity of a zero is the power of the factor corresponding to that zero:
   - \( x = 0 \) comes from \( x^2 \), so the multiplicity is 2.
   - \( x = 1 \) comes from \( (x - 1)^3 \), so the multiplicity is 3.
   - \( x = -2 \) comes from \( (x + 2)^2 \), so the multiplicity is 2.

### Final Answer:
- The zeros are \( x = 0 \), \( x = 1 \), and \( x = -2 \).
- The multiplicities are:
   - \( x = 0 \) (multiplicity 2),
   - \( x = 1 \) (multiplicity 3),
   - \( x = -2 \) (multiplicity 2).

Would you like further details or have any other questions? Here are some related questions you might find interesting:

1. What is the significance of multiplicity in determining the shape of the graph near each zero?
2. How does the negative coefficient \(-4\) affect the overall graph of the function?
3. Can a zero with an even multiplicity ever cross the x-axis?
4. How would you find the turning points of this polynomial function?
5. How do you determine the end behavior of a polynomial function like this one?

**Tip:** A zero with an even multiplicity (like \(x = 0\) and \(x = -2\)) will cause the graph to touch the x-axis and bounce off, whereas a zero with odd multiplicity (like \(x = 1\)) will cause the graph to cross the x-axis.

Find the zeros of the function z(x) = -4x^2 (x - 1)^3 (x + 2)^2 and state their multiplicities.

Learn how to find the zeros of the polynomial function z(x) = -4x^2 (x - 1)^3 (x + 2)^2 and determine their multiplicities. Understand key algebraic concepts like the Zero Factor Theorem and the role of multiplicity in shaping the graph. Suitable for Grades 10-12.

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