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:
-
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)
-
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)
-
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)
-
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.