Zero(s) where the graph crosses the -axis:

Zero(s) where the graph touches, but does not cross the -axis:

To summarize the behavior at each zero for the polynomial function \( f(x) = x^2 (x - 2)^2 (x + 1) \):

### Zeros where the graph crosses the x-axis:
- \( x = -1 \) (odd multiplicity)

### Zeros where the graph touches, but does not cross the x-axis:
- \( x = 0 \) (even multiplicity)
- \( x = 2 \) (even multiplicity)

Would you like more details or have any questions about this explanation?

Here are some additional questions you might have:
1. What does it mean for a zero to have even or odd multiplicity?
2. How do you find the zeros of a polynomial function?
3. Why does the graph of a polynomial touch but not cross the x-axis at zeros with even multiplicity?
4. How do you use the multiplicity of a zero to sketch the graph of a polynomial?
5. Can you show an example of a polynomial with zeros of different multiplicities?

Tip: Always look at the multiplicity of the zeros to understand how the graph behaves at each zero. This helps in sketching the graph accurately.

Explore polynomial functions and their zeroes, focusing on how the graph crosses or touches the x-axis. Learn about multiplicity and its impact on the graph's behavior. Suitable for high school students studying algebra and polynomial functions.

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