Question

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

a. use the leading coefficient test to determine the graphs end behavior.
which of the following is the correct statement about the end behavior of the given function?

a. the graph rises to the left and falls to the right.

b. the graph rises to the left and to the right.

c. the graph falls to the left and rises to the right.

d. the graph falls to the left and to the right.

Explanation:

Step1: Find the leading term

First, expand the polynomial to find the leading term. The polynomial is \( f(x) = x^2(x - 3)^3(x + 5) \). When we multiply the leading terms of each factor: the leading term of \( x^2 \) is \( x^2 \), of \( (x - 3)^3 \) is \( x^3 \), and of \( (x + 5) \) is \( x \). Multiplying these together: \( x^2 \cdot x^3 \cdot x = x^{2 + 3 + 1} = x^6 \). So the leading term is \( x^6 \), and the leading coefficient is \( 1 \) (which is positive), and the degree is \( 6 \) (even).

Step2: Apply Leading Coefficient Test

For the Leading Coefficient Test:

• If the degree of the polynomial is even:
• If the leading coefficient is positive, the graph rises to the left and to the right.
• If the leading coefficient is negative, the graph falls to the left and to the right.
• If the degree of the polynomial is odd:
• If the leading coefficient is positive, the graph falls to the left and rises to the right.
• If the leading coefficient is negative, the graph rises to the left and falls to the right.

Here, degree \( 6 \) (even) and leading coefficient \( 1 \) (positive). So the graph rises to the left and to the right.

Answer:

B. The graph rises to the left and to the right.