Refer to the polynomial function h(x) that you wrote in Question 1 of Finding

Question

the Zeros of a Polynomial Function and find the zeros you located. Choose x�values 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!