Question
Sketch f(x)=2x4−6x3−8x2. How does the graph behave at the zeros of the polynomial?(1 point) Responses The graph crosses the x-axis at 4 and −1. The graph crosses the x -axis at 4 and negative 1 . The graph crosses the x-axis at 4 and −1, and touches the x-axis and turns around at 0. The graph crosses the x -axis at 4 and negative 1 , and touches the x -axis and turns around at 0. The graph crosses the x-axis and turns around at 4 and −1. The graph crosses the x -axis and turns around at 4 and negative 1 . The graph crosses the x-axis and turns around at 4 and −1, and touches the x-axis at 0. The graph crosses the x -axis and turns around at 4 and negative 1 , and touches the x -axis at 0.
Answers
GPT-4o mini
To analyze the behavior of the polynomial \( f(x) = 2x^4 - 6x^3 - 8x^2 \) at its zeros, we first need to find its factors and identify the zeros.
1. **Finding the zeros:**
Factor out the common term from the polynomial:
\[
f(x) = 2x^2(x^2 - 3x - 4)
\]
Now, we can factor the quadratic:
\[
x^2 - 3x - 4 = (x - 4)(x + 1)
\]
Therefore, the complete factorization of \( f(x) \) is:
\[
f(x) = 2x^2(x - 4)(x + 1)
\]
The zeros of the polynomial are:
- \( x = 0 \) (with a multiplicity of 2),
- \( x = 4 \) (with a multiplicity of 1),
- \( x = -1 \) (with a multiplicity of 1).
2. **Behavior at the zeros:**
- At \( x = 0\), since the multiplicity is even (2), the graph touches the x-axis and turns around.
- At \( x = 4\), since the multiplicity is odd (1), the graph crosses the x-axis.
- At \( x = -1\), since the multiplicity is also odd (1), the graph crosses the x-axis here as well.
3. **Conclusion:**
The graph crosses the x-axis at \( x = 4 \) and \( x = -1 \), and it touches the x-axis and turns around at \( x = 0 \).
This matches the response:
**The graph crosses the x-axis at 4 and -1, and touches the x-axis and turns around at 0.**
1. **Finding the zeros:**
Factor out the common term from the polynomial:
\[
f(x) = 2x^2(x^2 - 3x - 4)
\]
Now, we can factor the quadratic:
\[
x^2 - 3x - 4 = (x - 4)(x + 1)
\]
Therefore, the complete factorization of \( f(x) \) is:
\[
f(x) = 2x^2(x - 4)(x + 1)
\]
The zeros of the polynomial are:
- \( x = 0 \) (with a multiplicity of 2),
- \( x = 4 \) (with a multiplicity of 1),
- \( x = -1 \) (with a multiplicity of 1).
2. **Behavior at the zeros:**
- At \( x = 0\), since the multiplicity is even (2), the graph touches the x-axis and turns around.
- At \( x = 4\), since the multiplicity is odd (1), the graph crosses the x-axis.
- At \( x = -1\), since the multiplicity is also odd (1), the graph crosses the x-axis here as well.
3. **Conclusion:**
The graph crosses the x-axis at \( x = 4 \) and \( x = -1 \), and it touches the x-axis and turns around at \( x = 0 \).
This matches the response:
**The graph crosses the x-axis at 4 and -1, and touches the x-axis and turns around at 0.**