Question

answer parts (e) for the question shown below. f(x)=x³ + 2x² - x - 2 a. use the leading - coefficient test to determine the graphs end behavior. which statement describes the behavior at the ends of f(x)=x³ + 2x² - x - 2? a. the graph falls to the left and rises to the right. 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= (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.

Explanation:

Step1: Identify the leading - term

The leading - term of the polynomial function \(f(x)=x^{3}+2x^{2}-x - 2\) is \(x^{3}\), where the degree \(n = 3\) (odd) and the leading coefficient \(a=1\) (positive).

Step2: Apply the leading - coefficient test

For a polynomial \(y = a_{n}x^{n}+a_{n - 1}x^{n - 1}+\cdots+a_{0}\), when \(n\) is odd and \(a_{n}>0\), as \(x\to-\infty\), \(y\to-\infty\) (the graph falls to the left) and as \(x\to+\infty\), \(y\to+\infty\) (the graph rises to the right). So the answer to part a is A.

Step3: Factor the polynomial to find x - intercepts

We can factor \(f(x)=x^{3}+2x^{2}-x - 2\) by grouping. \(f(x)=x^{2}(x + 2)-(x + 2)=(x + 2)(x^{2}-1)=(x + 2)(x + 1)(x - 1)\). Set \(f(x)=0\), then \(x=-2,-1,1\).

Step4: Determine the behavior at x - intercepts

If the multiplicity of a zero \(r\) of a polynomial \(y = f(x)\) is odd, the graph of \(y = f(x)\) crosses the \(x\) - axis at \(x = r\). Since the zeros \(x=-2,x=-1,x = 1\) all have multiplicity 1 (odd), the graph crosses the \(x\) - axis at \(x=-2,x=-1,x = 1\).

Answer:

a. A. The graph falls to the left and rises to the right.
b. \(x=-2,-1,1\); A. \(x=-2,-1,1\)