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

Question

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).