Question

Refer to the polynomial function h(x) that you wrote in Question 1 of Finding
the Zeros of a Polynomial Function and find the zeros you located. Choose xvalues and determine what the graph of h(x) will look like between its zeros. In
particular, make sure to determine whether the graph will be above the x-axis
or below the x-axis in each region.

Answer 1

To assist you accurately, I'll need a bit more context since you referenced a "Question 1" and a specific polynomial function \( h(x) \) which I don't have access to. However, I can provide you with a general guide on how to find the zeros of a polynomial function, analyze these zeros, and determine the behavior of the graph of \( h(x) \) between its zeros.

1. **Finding the Zeros:** Start by identifying the polynomial \( h(x) \). For example, if \( h(x) = x^3 - 4x \), to find the zeros, set \( h(x) = 0 \):
\[
x^3 - 4x = 0 \implies x(x^2 - 4) = 0 \implies x(x - 2)(x + 2) = 0
\]
Thus, the zeros are \( x = -2, 0, 2 \).

2. **Choosing x-values:** Once you have the zeros, select x-values that lie in the intervals defined by these zeros. For the example above, the intervals would be:
- \( (-\infty, -2) \)
- \( (-2, 0) \)
- \( (0, 2) \)
- \( (2, \infty) \)

3. **Determine the graph's behavior in each region:** Choose values from each interval to evaluate \( h(x) \):
- **For \( (-\infty, -2) \):** Choose \( x = -3 \):
\[
h(-3) = (-3)^3 - 4(-3) = -27 + 12 = -15
\]
So, \( h(-3) < 0 \) (the graph is below the x-axis).

- **For \( (-2, 0) \):** Choose \( x = -1 \):
\[
h(-1) = (-1)^3 - 4(-1) = -1 + 4 = 3
\]
So, \( h(-1) > 0 \) (the graph is above the x-axis).

- **For \( (0, 2) \):** Choose \( x = 1 \):
\[
h(1) = (1)^3 - 4(1) = 1 - 4 = -3
\]
So, \( h(1) < 0 \) (the graph is below the x-axis).

- **For \( (2, \infty) \):** Choose \( x = 3 \):
\[
h(3) = (3)^3 - 4(3) = 27 - 12 = 15
\]
So, \( h(3) > 0 \) (the graph is above the x-axis).

4. **Conclusion on Intervals:** Based on the evaluations, the behavior of the graph between the zeros is:
- In \( (-\infty, -2) \): Below the x-axis
- In \( (-2, 0) \): Above the x-axis
- In \( (0, 2) \): Below the x-axis
- In \( (2, \infty) \): Above the x-axis

The graph will cross the x-axis at each zero, and the behavior between the zeros alternates based on the polynomial’s degree and the number of times that zero is repeated (odd/even multiplicity).

If you provide the specific polynomial function \( h(x) \) mentioned, I can tailor the explanation with that function in mind!