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
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

Answer 1

Here’s the completed table based on the given polynomial \( P(x) = x^2(3x+6)^3(2x-4)(x+15)^4 \):

| 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 |

### Explanation:
- **Zeros**: The zeros of each factor are determined by setting each factor equal to zero.
- **Multiplicity**: The exponent of each linear factor indicates its multiplicity.
- **Even/Odd**: If the multiplicity is even, the factor is classified as "Even," and if it's odd, it's classified as "Odd."
- **Behavior at the x-intercept**:
- If the factor has an even multiplicity, the polynomial touches the x-axis and turns back (does not cross).
- If the factor has an odd multiplicity, the polynomial crosses the x-axis (changes sign).