Question

for the polynomial function (f(x) = -x^2(x - 3)(x + 5)), answer parts a through e.

a. use the leading coefficient test to determine the graphs end behavior.

a. the graph of (f(x)) rises to the left and falls to the right.
b. the graph of (f(x)) falls to the left and rises to the right.
c. the graph of (f(x)) falls to the left and falls to the right.
d. the graph of (f(x)) rises to the left and rises to the right.

b. find the x-intercept(s).

the x-intercept(s) is/are .
(type an integer or a decimal. use a comma to separate answers as needed.)

Explanation:

Determine the degree and leading coefficient

Using the Degree of Polynomial and Polynomial Functions knowledge points

$$
LATEXBLOCK0
$$

Apply the Leading Coefficient Test

Using the Leading Coefficient Test and End Behavior of Polynomials knowledge points

$$
LATEXBLOCK1
$$

Find the x-intercepts

To find the \(x\)-intercepts, set the function \(f(x) = 0\):

$$ -x^2(x-3)(x+5) = 0 $$

Solving for \(x\) gives:

$$
LATEXBLOCK2
$$

Thus, the \(x\)-intercepts are \(0, 3, -5\).

Answer:

Question a

• (A) The graph of f(x) rises to the left and falls to the right.
• (B) The graph of f(x) falls to the left and rises to the right.
• (C) The graph of f(x) falls to the left and falls to the right. (Correct answer)
• (D) The graph of f(x) rises to the left and rises to the right.

Question b

The x-intercept(s) is/are <blank>\(0, 3, -5\)</blank>