Question

for the polynomial (-x^4 - 4x^3 + 16x^2 + 64x), find the following:

a) the end behavior of the graph using the leading coefficient test.

• falls left and rises right
• rises left and falls right
• rises left and right
• falls left and right

b) all x-intercepts. separate each intercept with a comma.

c) the behavior of the graph at all x-intercepts. if there is not an x-intercept with the given behavior, state \dne\ in the box next to the behavior.

x-intercepts where the graph crosses the x-axis

x-intercepts where the graph touches the x-axis and turns

d) the y-intercept.

Explanation:

Step1: Identify leading term

The leading term is \(-x^4\).

Step2: Determine end behavior

Since degree is even and leading coefficient is negative:

$$\text{Falls left and right}$$

Step3: Factor the polynomial

Factor out \(-x\):

$$-x(x^3 + 4x^2 - 16x - 64)$$

Step4: Factor by grouping

Group terms inside:

$$-x[x^2(x + 4) - 16(x + 4)] = -x(x^2 - 16)(x + 4)$$

Step5: Fully factor expression

Apply difference of squares:

$$-x(x - 4)(x + 4)^2$$

Step6: Find x-intercepts

Set each factor to zero:

$$x = 0, \quad x = 4, \quad x = -4$$

Step7: Determine crossing behavior

Odd multiplicity roots cross x-axis:

$$x = 0, \quad x = 4$$

Step8: Determine touching behavior

Even multiplicity roots touch and turn:

$$x = -4$$

Step9: Find y-intercept

Evaluate at \(x = 0\):

$$y = 0$$

Answer:

a) Falls left and right
b) -4, 0, 4
c) x-intercepts where the graph crosses the x-axis: -4, 0, 4
x-intercepts where the graph touches the x-axis and turns: DNE
d) 0