Question

for the polynomial function f(x)= - 3x^4 - 9x^3, answer the parts a through e.
a. use the leading coefficient test to determine the graphs end behavior.
a. the graph of f(x) falls 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) rises 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 - intercepts. state whether the graph crosses the x - axis, or touches the x - axis and turns around, at each intercept.
the x - intercept(s) is/are - 3,0.
(type an integer or a decimal. use a comma to separate answers as needed. type each answer only once.)
at which x - intercept(s) does the graph cross the x - axis? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the x - intercept(s) at which the graph crosses the x - axis is/are
(type an integer or a decimal. use a comma to separate answers as needed. type each answer only once.)
b. there are no x - intercepts at which the graph crosses the x - axis.

Explanation:

Step1: Determine leading - coefficient and degree

The polynomial function is $f(x)=-3x^{4}-9x^{3}$, the leading - term is $-3x^{4}$, the leading coefficient $a=-3<0$ and the degree $n = 4$ (even). According to the leading - coefficient test, when $a<0$ and $n$ is even, the graph of $y = f(x)$ falls to the left and falls to the right.

Step2: Find x - intercepts

Set $f(x)=0$, so $-3x^{4}-9x^{3}=0$. Factor out $-3x^{3}$: $-3x^{3}(x + 3)=0$. Then, by the zero - product property, $-3x^{3}=0$ gives $x = 0$ and $x+3=0$ gives $x=-3$. The x - intercepts are $x = 0$ and $x=-3$.
For $x = 0$, the factor is $x^{3}$ (odd power), so the graph crosses the x - axis at $x = 0$. For $x=-3$, the factor is $(x + 3)$ (odd power), so the graph crosses the x - axis at $x=-3$.

Answer:

a. A. The graph of f(x) falls to the left and falls to the right.
b. The x - intercept(s) is/are $-3,0$.
A. The x - intercept(s) at which the graph crosses the x - axis is/are $-3,0$.