Question

answer parts (a)-(e) for the function shown below. f(x)=x^{3}+2x^{2}-x - 2 (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= b. there are no x - intercepts at which the graph touches the x - axis and turns around. c. find the y - intercept. the y - intercept is y=-2. (type an integer or a decimal.) d. determine whether the graph has y - axis symmetry, origin symmetry, or neither. choose the correct answer below. a. origin symmetry b. y - axis symmetry c. neither

Explanation:

Step1: Find x - intercepts

Set \(f(x)=x^{3}+2x^{2}-x - 2 = 0\). Factor by grouping: \(x^{2}(x + 2)-(x + 2)=0\), then \((x + 2)(x^{2}-1)=0\), and further \((x + 2)(x - 1)(x+1)=0\). Solving gives \(x=-2,-1,1\). These are the points where the graph crosses the x - axis.

Step2: Check for graph touching x - axis

For a graph of a polynomial \(y = f(x)\) to touch and turn around at an x - intercept \(a\), the factor \((x - a)\) must have an even multiplicity. The factored form \(f(x)=(x + 2)(x - 1)(x + 1)\) has all factors with multiplicity 1, so there are no x - intercepts where the graph touches and turns around.

Step3: Find y - intercept

Set \(x = 0\) in \(f(x)\). Then \(f(0)=0^{3}+2\times0^{2}-0 - 2=-2\).

Step4: Check for symmetries

For y - axis symmetry, check if \(f(-x)=f(x)\). \(f(-x)=(-x)^{3}+2(-x)^{2}-(-x)-2=-x^{3}+2x^{2}+x - 2\), and \(f(x)=x^{3}+2x^{2}-x - 2\), so \(f(-x)
eq f(x)\). For origin symmetry, check if \(f(-x)=-f(x)\). \(-f(x)=-x^{3}-2x^{2}+x + 2\), and \(f(-x)=-x^{3}+2x^{2}+x - 2\), so \(f(-x)
eq -f(x)\). The graph has neither y - axis nor origin symmetry.

Answer:

a. A. \(x=-2,-1,1\)
b. B. There are no x - intercepts at which the graph touches the x - axis and turns around.
c. \(y=-2\)
d. C. neither