Question

Identify the multiplicities of the linear factors of f(x)=(x+4)(x−1)2(x+3)5
. Then find the zeros of f(x). State whether the function crosses the x-axis or bounces off the x-axis at each of the zeros.

You must show thorough, handwritten work. You may only use scratch paper, a writing utensil, and a calculator that does not have a computer algebra system.

Answer 1

To analyze the polynomial function \( f(x) = (x+4)(x-1)^2(x+3)^5 \), we will identify the linear factors along with their multiplicities and determine their respective zeros.

### Step 1: Identify the linear factors and their multiplicities

1. **Factor \( (x + 4) \)**:
- Zero: \( x = -4 \)
- Multiplicity: 1 (because the factor appears to the first power)

2. **Factor \( (x - 1)^2 \)**:
- Zero: \( x = 1 \)
- Multiplicity: 2 (because the factor appears to the second power)

3. **Factor \( (x + 3)^5 \)**:
- Zero: \( x = -3 \)
- Multiplicity: 5 (because the factor appears to the fifth power)

### Step 2: List the zeros of \( f(x) \)

The zeros of the function \( f(x) \) are:

- \( x = -4 \) with multiplicity 1
- \( x = 1 \) with multiplicity 2
- \( x = -3 \) with multiplicity 5

### Step 3: Determine whether \( f(x) \) crosses or bounces at the x-axis

The behavior of the function at the x-axis is determined by the multiplicity of the zeros:

- A zero with **odd multiplicity** means the graph **crosses** the x-axis.
- A zero with **even multiplicity** means the graph **bounces** off the x-axis.

Analysis of the multiplicities:
1. **At \( x = -4 \)** (multiplicity 1, odd): The function crosses the x-axis.
2. **At \( x = 1 \)** (multiplicity 2, even): The function bounces off the x-axis.
3. **At \( x = -3 \)** (multiplicity 5, odd): The function crosses the x-axis.

### Summary of the Zeros and Behavior

- \( x = -4 \): Crosses the x-axis
- \( x = 1 \): Bounces off the x-axis
- \( x = -3 \): Crosses the x-axis

### Conclusion

- Zeros: \( -4, 1, -3 \)
- Behavior:
- \( x = -4 \): crosses
- \( x = 1 \): bounces
- \( x = -3 \): crosses

This input should provide you with the required analysis of the function \( f(x) = (x+4)(x−1)^2(x+3)^5 \).