Question

for the polynomial (f(x) = -\frac{1}{2}x^3(x + 11)^2(x - 9)^4) perform the following tasks:

a) identify the x-intercepts from smallest to largest and the multiplicity of each intercept. if there are more boxes than zeros, enter \dne\ in the boxes that are not needed.

smallest zero: multiplicity

second zero: multiplicity

third zero: multiplicity

b) determine the intercepts where the graph crosses the x-axis and where it touches the x-axis and turns. if there is no answer for one of the behaviors, enter \dne\ in the boxes that are not needed.

zeros where the graph crosses the x-axis:

zeros where the graph touches the x-axis and turns:

Explanation:

Step1: Find the zeros of the polynomial

$$f(x) = -\frac{1}{2}x^3(x + 11)^2(x - 9)^4 = 0 \implies x = 0, -11, 9$$

Step2: Identify multiplicities of each zero

$$x = -11 \text{ has multiplicity } 2, \quad x = 0 \text{ has multiplicity } 3, \quad x = 9 \text{ has multiplicity } 4$$

Step3: Order zeros from smallest to largest

$$\text{Smallest: } -11, \quad \text{Second: } 0, \quad \text{Third: } 9$$

Step4: Determine behavior at each zero

$$\text{Odd multiplicity (crosses): } 0, \quad \text{Even multiplicity (touches and turns): } -11, 9$$

Answer:

A)
smallest zero: \(-11\), multiplicity: \(2\)
second zero: \(0\), multiplicity: \(3\)
third zero: \(9\), multiplicity: \(4\)

B)
zeros where the graph crosses the x-axis: \(0\)
zeros where the graph touches the x-axis and turns: \(-11, 9\)