Question

Fill in the rest of the table to describe the polynomial at it’s x-intercepts. x2(3x+6)3(2x-4)(x+15)4


Linear Factor
Zeros
Multiplicity
Even/Odd
Behavior at the x-intercept
x2


2




(3x+6)3




Odd


(2x-4)
2






(x+15)4






Touch, turn back

Answer 1

To fill in the rest of the table regarding the polynomial \( x^2(3x+6)^3(2x-4)(x+15)^4 \), we need to analyze each factor:

### Analyzing Each Factor:
1. **For \( x^2 \)**:
- **Linear Factor**: \( x^2 \)
- **Zeros**: \( 0 \)
- **Multiplicity**: \( 2 \) (since it is \( x^2 \))
- **Even/Odd**: Even (multiplicity of \( 2 \))
- **Behavior at the x-intercept**: Touch, turn back (since it's even)

2. **For \( (3x+6)^3 \)**:
- **Linear Factor**: \( 3x+6 \)
- **Zeros**: \( -2 \) (Set \( 3x+6 = 0 \) → \( 3x = -6 → x = -2 \))
- **Multiplicity**: \( 3 \) (since it is cubed)
- **Even/Odd**: Odd (since the multiplicity is \( 3 \))
- **Behavior at the x-intercept**: Cross (since it's odd)

3. **For \( (2x-4) \)**:
- **Linear Factor**: \( 2x-4 \)
- **Zeros**: \( 2 \) (Set \( 2x-4 = 0 \) → \( 2x = 4 → x = 2 \))
- **Multiplicity**: \( 1 \) (since it is linear)
- **Even/Odd**: Odd (since the multiplicity is \( 1 \))
- **Behavior at the x-intercept**: Cross (since it's odd)

4. **For \( (x+15)^4 \)**:
- **Linear Factor**: \( x+15 \)
- **Zeros**: \( -15 \) (Set \( x+15 = 0 \) → \( x = -15 \))
- **Multiplicity**: \( 4 \) (since it is raised to the 4th power)
- **Even/Odd**: Even (multiplicity of \( 4 \))
- **Behavior at the x-intercept**: Touch, turn back (since it's even)

### Completed Table:

| Linear Factor | Zeros | Multiplicity | Even/Odd | Behavior at the x-intercept |
|------------------|-------|--------------|----------|---------------------------------|
| \( x^2 \) | \( 0 \) | \( 2 \) | Even | Touch, turn back |
| \( (3x+6)^3 \) | \( -2 \) | \( 3 \) | Odd | Cross |
| \( (2x-4) \) | \( 2 \) | \( 1 \) | Odd | Cross |
| \( (x+15)^4 \) | \( -15 \) | \( 4 \) | Even | Touch, turn back |

This table provides the needed information about the polynomial at its x-intercepts.