Question

answer parts (a)-(o) for the function shown below. f(x)=x^{3}+2x^{2}-x - 2

b. the graph rises to the left and falls to the right.
c. the graph falls to the left and to the right.
d. the graph rises to the left and 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. what are the x - intercepts?
x = - 2,-1,1 (type an integer or a decimal. use a comma to separate answers as needed.)
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. x = - 2,-1,1 (type an integer or a decimal. use a comma to separate answers as needed.)
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. x =
(type an integer or a decimal. use a comma to separate answers as needed.)
b. there are no x - intercepts at which the graph touches the x - axis and turns around.

Explanation:

Step1: Analyze end - behavior of polynomial

For the polynomial function $f(x)=x^{3}+2x^{2}-x - 2$, the leading term is $x^{3}$ with a positive leading coefficient ($a = 1$) and an odd degree ($n=3$). As $x\to-\infty$, $y\to-\infty$ and as $x\to+\infty$, $y\to+\infty$. So the graph falls to the left and rises to the right.

Step2: Find x - intercepts

Factor the polynomial $f(x)=x^{3}+2x^{2}-x - 2=x^{2}(x + 2)-(x + 2)=(x + 2)(x^{2}-1)=(x + 2)(x + 1)(x - 1)$. Set $f(x)=0$, then $(x + 2)(x + 1)(x - 1)=0$. Solving gives $x=-2,-1,1$.

Step3: Determine crossing or touching at x - intercepts

For a factor $(x - c)$ of a polynomial, if the multiplicity of the root $c$ is odd, the graph crosses the $x$-axis at $x = c$. Since the roots $x=-2,-1,1$ all have multiplicity 1 (odd), the graph crosses the $x$-axis at $x=-2,-1,1$. And there are no roots with even multiplicity, so the graph does not touch and turn around at any $x$-intercepts.

Answer:

a. No correct option shown in provided text (graph falls to the left and rises to the right).
b. x - intercepts: $x=-2,-1,1$
Graph crosses the $x$-axis at: $x=-2,-1,1$
Graph touches and turns around at: B. There are no x - intercepts at which the graph touches the x - axis and turns around.