Question
Sketch f(x)=2x^4-6x^3-8x^2
. How does the graph behave at the zeros of the polynomial?
A
The graph crosses the x-axis at 4 and −1, and touches the x-axis and turns around at 0.
B
The graph crosses the x-axis and turns around at 4 and −1, and touches the x-axis at 0.
C
The graph crosses the x-axis at 4 and −1.
D
The graph crosses the x-axis and turns around at 4 and −1
. How does the graph behave at the zeros of the polynomial?
A
The graph crosses the x-axis at 4 and −1, and touches the x-axis and turns around at 0.
B
The graph crosses the x-axis and turns around at 4 and −1, and touches the x-axis at 0.
C
The graph crosses the x-axis at 4 and −1.
D
The graph crosses the x-axis and turns around at 4 and −1
Answers
GPT-4o mini
To determine the behavior of the graph of the polynomial \( f(x) = 2x^4 - 6x^3 - 8x^2 \) at its zeros, we first need to find the zeros of the function.
To do this, we can factor the polynomial:
1. **Factor out the common term:**
\[
f(x) = 2x^2 (x^2 - 3x - 4)
\]
2. **Factor the quadratic term:**
To factor \( x^2 - 3x - 4 \), we look for two numbers that multiply to \(-4\) and add to \(-3\). These numbers are \(-4\) and \(1\).
\[
x^2 - 3x - 4 = (x - 4)(x + 1)
\]
Putting it all together:
\[
f(x) = 2x^2 (x - 4)(x + 1)
\]
Now we can identify the zeros of the polynomial:
- \( x = 0 \) (with a multiplicity of 2)
- \( x = 4 \) (with multiplicity 1)
- \( x = -1 \) (with multiplicity 1)
### Analysis of Behavior at the Zeros:
- **At \( x = 0 \):** Since the multiplicity is 2 (even), the graph will touch the x-axis and turn around.
- **At \( x = 4 \):** Since the multiplicity is 1 (odd), the graph will cross the x-axis.
- **At \( x = -1 \):** Since the multiplicity is 1 (odd), the graph will also cross the x-axis.
### Conclusion:
With this information, we can summarize the behavior of the graph near the zeros:
- The graph **touches** the x-axis at \( x = 0 \) and **turns around**.
- The graph **crosses** the x-axis at \( x = 4 \) and \( x = -1 \).
Based on these observations, the correct answer is:
**A**: The graph crosses the x-axis at \( 4 \) and \( -1 \), and touches the x-axis and turns around at \( 0 \).
To do this, we can factor the polynomial:
1. **Factor out the common term:**
\[
f(x) = 2x^2 (x^2 - 3x - 4)
\]
2. **Factor the quadratic term:**
To factor \( x^2 - 3x - 4 \), we look for two numbers that multiply to \(-4\) and add to \(-3\). These numbers are \(-4\) and \(1\).
\[
x^2 - 3x - 4 = (x - 4)(x + 1)
\]
Putting it all together:
\[
f(x) = 2x^2 (x - 4)(x + 1)
\]
Now we can identify the zeros of the polynomial:
- \( x = 0 \) (with a multiplicity of 2)
- \( x = 4 \) (with multiplicity 1)
- \( x = -1 \) (with multiplicity 1)
### Analysis of Behavior at the Zeros:
- **At \( x = 0 \):** Since the multiplicity is 2 (even), the graph will touch the x-axis and turn around.
- **At \( x = 4 \):** Since the multiplicity is 1 (odd), the graph will cross the x-axis.
- **At \( x = -1 \):** Since the multiplicity is 1 (odd), the graph will also cross the x-axis.
### Conclusion:
With this information, we can summarize the behavior of the graph near the zeros:
- The graph **touches** the x-axis at \( x = 0 \) and **turns around**.
- The graph **crosses** the x-axis at \( x = 4 \) and \( x = -1 \).
Based on these observations, the correct answer is:
**A**: The graph crosses the x-axis at \( 4 \) and \( -1 \), and touches the x-axis and turns around at \( 0 \).