Question

for the polynomial function f(x) = -2x³(x - 1)²(x + 3), 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 rises 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 falls to the right.

Explanation:

Step1: Determine the degree of the polynomial

First, expand the polynomial to find its degree. The polynomial is \( f(x) = -2x^3(x - 1)^2(x + 3) \). When we multiply out the factors, the highest power of \( x \) (the degree) is found by adding the exponents of \( x \) from each factor. The exponents are 3 (from \( x^3 \)), 2 (from \( (x - 1)^2 \)), and 1 (from \( x + 3 \)). So the degree \( n = 3 + 2 + 1 = 6 \), which is even.

Step2: Determine the leading coefficient

The leading coefficient is the coefficient of the term with the highest degree. When we expand the polynomial, the leading term comes from multiplying the leading terms of each factor: \( -2x^3 \cdot x^2 \cdot x = -2x^6 \). So the leading coefficient \( a_n = -2 \), which is negative.

Step3: Apply the Leading Coefficient Test

For a polynomial with degree \( n \) (even or odd) and leading coefficient \( a_n \):

• If \( n \) is even and \( a_n > 0 \), the graph rises to the left and rises to the right.
• If \( n \) is even and \( a_n < 0 \), the graph falls to the left and falls to the right.
• If \( n \) is odd and \( a_n > 0 \), the graph falls to the left and rises to the right.
• If \( n \) is odd and \( a_n < 0 \), the graph rises to the left and falls to the right.

Here, \( n = 6 \) (even) and \( a_n = -2 \) (negative). So the graph falls to the left and falls to the right.

Answer:

C. The graph of \( f(x) \) falls to the left and falls to the right.