Question
Question content area top left
Part 1
Answer parts (a)dash(e) for the function shown below.
f(x)equals x cubed plus 3 x squared minus x minus 3
Question content area bottom left
Part 1
a. Use the leading coefficient test to determine the graph's end behavior. Which statement describes the behavior at the ends of f(x)equals x cubed plus 3 x squared minus x minus 3?
A.
The graph falls to the left and rises to the right.
B.
The graph rises to the left and to the right.
C.
The graph rises to the left and falls to the right.
D.
The graph falls to the left and to the right.
.
.
.
Question content area right
Part 1
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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?
xequals
enter your response here
(Type an integer or a decimal. Use a comma to separate answers as needed.)
Part 1
Answer parts (a)dash(e) for the function shown below.
f(x)equals x cubed plus 3 x squared minus x minus 3
Question content area bottom left
Part 1
a. Use the leading coefficient test to determine the graph's end behavior. Which statement describes the behavior at the ends of f(x)equals x cubed plus 3 x squared minus x minus 3?
A.
The graph falls to the left and rises to the right.
B.
The graph rises to the left and to the right.
C.
The graph rises to the left and falls to the right.
D.
The graph falls to the left and to the right.
.
.
.
Question content area right
Part 1
-10
-8
-6
-4
-2
2
4
6
8
10
-10
-8
-6
-4
-2
2
4
6
8
10
x
y
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?
xequals
enter your response here
(Type an integer or a decimal. Use a comma to separate answers as needed.)
Answers
GPT-4o mini
To answer the questions for the function \( f(x) = x^3 + 3x^2 - x - 3 \), we can analyze it step by step:
### Part (a): Leading Coefficient Test
The leading term of the polynomial \( f(x) \) is \( x^3 \). The leading coefficient (the coefficient of the leading term) is positive (1).
Since the degree of the polynomial (3) is odd and the leading coefficient is positive, we can determine the end behavior of the graph:
- As \( x \to -\infty \) (to the left), \( f(x) \to -\infty \) (the graph falls).
- As \( x \to +\infty \) (to the right), \( f(x) \to +\infty \) (the graph rises).
Thus, the correct statement describing the behavior at the ends of the graph is:
**A. The graph falls to the left and rises to the right.**
### Part (b): Finding the x-intercepts
To find the x-intercepts, we need to solve the equation \( f(x) = 0 \):
\[
x^3 + 3x^2 - x - 3 = 0
\]
To find the rational roots, we can use the Rational Root Theorem, looking for factors of the constant term (-3) over the leading coefficient (1). The possible rational roots are \( \pm 1, \pm 3 \).
We can test these values to see if they are roots:
1. **Testing \( x = 1 \)**:
\[
f(1) = 1^3 + 3(1^2) - 1 - 3 = 1 + 3 - 1 - 3 = 0
\]
So, \( x = 1 \) is a root.
Now we can factor \( (x - 1) \) out from \( f(x) \). We can use synthetic division to divide \( f(x) \) by \( (x - 1) \).
Using synthetic division with \( 1 \):
- Coefficients are: \( 1, 3, -1, -3 \)
- Performing synthetic division:
```
1 | 1 3 -1 -3
| 1 4 3
---------------------
1 4 3 0
```
The resulting polynomial is \( x^2 + 4x + 3 \). Now we need to factor this further:
\[
x^2 + 4x + 3 = (x + 1)(x + 3)
\]
So,
\[
f(x) = (x - 1)(x + 1)(x + 3)
\]
Now we find the x-intercepts by setting each factor equal to zero:
1. \( x - 1 = 0 \) ⟹ \( x = 1 \)
2. \( x + 1 = 0 \) ⟹ \( x = -1 \)
3. \( x + 3 = 0 \) ⟹ \( x = -3 \)
The x-intercepts are \( x = 1, -1, -3 \).
### Determine if the graph crosses or touches the x-axis at each intercept:
- The factor \( (x - 1) \): Since this is a linear factor, the graph **crosses** the x-axis at \( x = 1 \).
- The factors \( (x + 1) \) and \( (x + 3) \) are also linear, meaning the graph will **cross** the x-axis at both \( x = -1 \) and \( x = -3 \).
### Final Response
**x-intercepts**: \( 1, -1, -3 \) (the graph crosses the x-axis at all of these points).
### Part (a): Leading Coefficient Test
The leading term of the polynomial \( f(x) \) is \( x^3 \). The leading coefficient (the coefficient of the leading term) is positive (1).
Since the degree of the polynomial (3) is odd and the leading coefficient is positive, we can determine the end behavior of the graph:
- As \( x \to -\infty \) (to the left), \( f(x) \to -\infty \) (the graph falls).
- As \( x \to +\infty \) (to the right), \( f(x) \to +\infty \) (the graph rises).
Thus, the correct statement describing the behavior at the ends of the graph is:
**A. The graph falls to the left and rises to the right.**
### Part (b): Finding the x-intercepts
To find the x-intercepts, we need to solve the equation \( f(x) = 0 \):
\[
x^3 + 3x^2 - x - 3 = 0
\]
To find the rational roots, we can use the Rational Root Theorem, looking for factors of the constant term (-3) over the leading coefficient (1). The possible rational roots are \( \pm 1, \pm 3 \).
We can test these values to see if they are roots:
1. **Testing \( x = 1 \)**:
\[
f(1) = 1^3 + 3(1^2) - 1 - 3 = 1 + 3 - 1 - 3 = 0
\]
So, \( x = 1 \) is a root.
Now we can factor \( (x - 1) \) out from \( f(x) \). We can use synthetic division to divide \( f(x) \) by \( (x - 1) \).
Using synthetic division with \( 1 \):
- Coefficients are: \( 1, 3, -1, -3 \)
- Performing synthetic division:
```
1 | 1 3 -1 -3
| 1 4 3
---------------------
1 4 3 0
```
The resulting polynomial is \( x^2 + 4x + 3 \). Now we need to factor this further:
\[
x^2 + 4x + 3 = (x + 1)(x + 3)
\]
So,
\[
f(x) = (x - 1)(x + 1)(x + 3)
\]
Now we find the x-intercepts by setting each factor equal to zero:
1. \( x - 1 = 0 \) ⟹ \( x = 1 \)
2. \( x + 1 = 0 \) ⟹ \( x = -1 \)
3. \( x + 3 = 0 \) ⟹ \( x = -3 \)
The x-intercepts are \( x = 1, -1, -3 \).
### Determine if the graph crosses or touches the x-axis at each intercept:
- The factor \( (x - 1) \): Since this is a linear factor, the graph **crosses** the x-axis at \( x = 1 \).
- The factors \( (x + 1) \) and \( (x + 3) \) are also linear, meaning the graph will **cross** the x-axis at both \( x = -1 \) and \( x = -3 \).
### Final Response
**x-intercepts**: \( 1, -1, -3 \) (the graph crosses the x-axis at all of these points).