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Let's address the given polynomial problem step by step.

### Part (a): Least Possible Number of Zeros

The zeros of a polynomial function \( p(x) \) are the values of \( x \) where \( p(x) = 0 \). From the table:

- \( p(1) = 0 \)

This means \( x = 1 \) is definitely a zero of the polynomial.

In terms of the behavior of the function, it changes sign between:
- \( p(0) = -1 \) and \( p(1) = 0 \)
- \( p(2) = 5 \) and \( p(3) = 0 \)

This implies that there could be more zeros, but since the problem asks for the **least possible** number of zeros, we are only guaranteed to have **1 zero**, which occurs at \( x = 1 \).

So, the least possible number of zeros is **1**.

### Part (b): Least Possible Number of Local Extrema

Local extrema (maxima or minima) occur where the derivative of the polynomial \( p'(x) = 0 \), and the function changes direction. We can analyze the function's behavior based on its values:

- \( p(x) \) goes from 2 to -1 (decreasing) between \( x = -3 \) and \( x = 0 \).
- \( p(x) \) increases from -1 to 5 between \( x = 0 \) and \( x = 2 \).
- \( p(x) \) decreases again from 5 to -7 between \( x = 2 \) and \( x = 4 \).

This suggests there are changes in direction:
- There is a local minimum between \( x = -3 \) and \( x = 2 \), possibly around \( x = 0 \).
- There is a local maximum between \( x = 0 \) and \( x = 2 \).

Thus, the function changes direction at least twice, indicating **at least two extrema**: one local minimum and one local maximum.

So, the least possible number of local extrema is **2**.

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### Do you want more details or have any questions?

### Here are some related questions:
1. What is the maximum possible number of zeros that \( p(x) \) could have based on the table?
2. How could you determine the degree of \( p(x) \) from the table data?
3. What would happen if the polynomial had more than two extrema?
4. Can you estimate the location of the extrema more precisely from the data provided?
5. What can the signs of \( p'(x) \) tell us about the increasing or decreasing nature of \( p(x) \)?

### Tip: When analyzing local extrema, always look for changes in the sign of the function values to infer turning points.

The function p(x) is a polynomial defined for -3 ≤ x ≤ 4. The table gives values of p(x) for select values of x. (a) What is the least possible number of zeros that p could have? (b) What is the least possible number of local extrema that p could have?

Explore a polynomial function defined over the interval -3 ≤ x ≤ 4. Determine the least possible number of zeros and local extrema based on provided values of p(x). This problem covers polynomial behavior and the concepts of zeros and extrema, ideal for Grades 11-12.

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