Question

find the x-intercepts of the polynomial function. state whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept.

(f(x) = -x^2(x + 2)(x^2 - 1))

a. 0, touches the x-axis and turns around;
2, crosses the x-axis;
-1, touches the x-axis and turns around;
1, touches the x-axis and turns around

b. 0, touches the x-axis and turns around;
-2, crosses the x-axis;
1, touches the x-axis and turns around

c. 0, crosses the x-axis;
-2, crosses the x-axis;
-1, crosses the x-axis;
1, crosses the x-axis

d. 0, touches the x-axis and turns around;
-2, crosses the x-axis;
-1, crosses the x-axis;
1, crosses the x-axis

Explanation:

Factor the polynomial completely

Using the Finding Zeros of Factored Polynomials knowledge point

$$
LATEXBLOCK0
$$

Find the x-intercepts

Using the Finding Zeros of Factored Polynomials knowledge point

$$
LATEXBLOCK1
$$

Determine the multiplicity of each zero

Using the Multiplicity of Zeros knowledge point

$$
LATEXBLOCK2
$$

Determine the graph behavior at each intercept

Using the Graph Behavior at Zeros knowledge point

$$
LATEXBLOCK3
$$

Answer:

• A. 0, touches the x-axis and turns around;

2, crosses the x-axis;

• 1, touches the x-axis and turns around;

1, touches the x-axis and turns around

• B. 0, touches the x-axis and turns around;
• 2, crosses the x-axis;

1, touches the x-axis and turns around

• C. 0, crosses the x-axis;
• 2, crosses the x-axis;
• 1, crosses the x-axis;

1, crosses the x-axis

• **D. 0, touches the x-axis and turns around,
• 2, crosses the x-axis;
• 1, crosses the x-axis;

1, crosses the x-axis (Correct answer)**