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:
- \( -3x^2 \) contributes a zero at \( x = 0 \) (double root, since it is squared).
- The term \( (5x + a) \) gives a zero when \( 5x + a = 0 \).
- Solving gives \( x = -\frac{a}{5} \).
- 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:
- Draw your axes on graph paper.
- Mark the point \( (0, 0) \).
- 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:
- Mark the points \( (0, 0) \), \( (-1, 0) \), and \( (2, 0) \) on your graph.
- Label this paper as "Graph 1".
- Indicate which zeros correspond to \( h(x) \).
Summary of Graph 1:
- Zeros:
- \( x = 0 \)
- \( x = -1 \)
- \( x = 2 \)
- 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!