Question

Directions: Use this worksheet to record your answers to the questions or
problems for the Graphs of Polynomials portfolio. When you are finished, save this
worksheet with your answers and submit it for a portfolio grade.
Where indicated, draw figures on a blank sheet of paper or on a sheet of graph paper.
Be sure to label each sheet so that your teacher knows which answer goes with which
question. You can scan these figures and submit them as individual documents with
your portfolio worksheet, or you can take pictures of them and insert the pictures on
the portfolio worksheet.
Finding the Zeros of a Polynomial Function
Question 1
A. Choose whole
h(x) =-3x^2(5x--------+)(x- )
Plot the zeros of h(x) on a coordinate plane on a piece of graph paper. Label
this paper Graph 1.
Note: You will complete Graph 1 throughout the portfolio. Do not worry about
scaling the y-axis at this time.

Answer 1

To find the zeros of the polynomial function \( h(x) = -3x^2(5x + a)(x - b) \), you need to set the function equal to zero and solve for \( x \).

### Step 1: Identify the factors of the polynomial
The function can be factored as follows:
1. \( -3x^2 \) contributes a zero at \( x = 0 \) (double root, since it is squared).
2. The term \( (5x + a) \) gives a zero when \( 5x + a = 0 \).
- Solving gives \( x = -\frac{a}{5} \).
3. The term \( (x - b) \) gives a zero when \( x - b = 0 \).
- Solving gives \( x = b \).

### Step 2: Gather the zeros
Thus, the zeros of the function \( h(x) \) are:
- \( x = 0 \) (with a multiplicity of 2)
- \( x = -\frac{a}{5} \)
- \( x = b \)

### Step 3: Plotting the zeros
To plot the zeros on a coordinate plane:
1. Draw your axes on graph paper.
2. Mark the point \( (0, 0) \).
3. Calculate and plot \( -\frac{a}{5} \) and \( b \) for specific values of \( a \) and \( b \). You can choose values for \( a \) and \( b \) that make sense for your problem.

### Example:
Assuming \( a = 5 \) and \( b = 2 \):
- \( \text{Zero 1: } x = 0 \)
- \( \text{Zero 2: } x = -\frac{5}{5} = -1 \)
- \( \text{Zero 3: } x = 2 \)

### Final Plot Instructions:
1. Mark the points \( (0, 0) \), \( (-1, 0) \), and \( (2, 0) \) on your graph.
2. Label this paper as "Graph 1".
3. Indicate which zeros correspond to \( h(x) \).

### Summary of Graph 1:
1. Zeros:
- \( x = 0 \)
- \( x = -1 \)
- \( x = 2 \)
2. Include these points and any additional relevant annotations.

Remember to save and submit your graph with the worksheet. If you have specific values for \( a \) and \( b \) you'd like to use, let me know, and I can help you customize the zeros based on that!