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 - 3,0.
(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.
at which x - intercept(s) does the graph touch the x - axis and turn around? 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 touches the x - axis and turns around 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 touches the x - axis and turns around.

Explanation:

Step1: Determine end - behavior using leading coefficient test

The polynomial function is \(f(x)=-3x^{4}-9x^{3}\), the degree \(n = 4\) (even) and the leading coefficient \(a=-3\) (negative). For a polynomial \(y = a x^{n}\), when \(n\) is even and \(a<0\), as \(x\to-\infty\), \(y\to-\infty\) and as \(x\to+\infty\), \(y\to-\infty\). So the graph of \(f(x)\) 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 \(x=-3\) and \(x = 0\).
For \(x=-3\), the factor is \((x + 3)\) with multiplicity 1. When the multiplicity of a zero is odd, the graph crosses the x - axis at that zero. For \(x = 0\), the factor is \(x^{3}\) with multiplicity 3 (odd), so the graph crosses the x - axis at \(x = 0\). There are no x - intercepts where the graph touches the x - axis and turns around.

Answer:

a. A. The graph of \(f(x)\) falls to the left and falls to the right.
b. The x - intercept(s) is/are \(-3,0\).
The x - intercept(s) at which the graph crosses the x - axis is/are \(-3,0\).
B. There are no x - intercepts at which the graph touches the x - axis and turns around.