Question
Find the zeros for the given polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero.
f(x)equals x cubed minus 22 x squared plus 121 x
Question content area bottom
Part 1
Determine the zero(s), if they exist.
The zero(s) is/are
enter your response here.
(Type integers or decimals. Use a comma to separate answers as needed.)
Determine the behavior of the function at each zero. Select the correct choice below and, if necessary, fill in the answer boxes within your choice.
A.
The graph touches the x-axis and turns around at all zeros.
B.
The graph crosses the x-axis at xequals
1. The graph touches the x-axis and turns around at xequals
2.
(Type integers or decimals. Simplify your answers. Use a comma to separate answers as needed.)
C.
The graph crosses the x-axis at all zeros.
f(x)equals x cubed minus 22 x squared plus 121 x
Question content area bottom
Part 1
Determine the zero(s), if they exist.
The zero(s) is/are
enter your response here.
(Type integers or decimals. Use a comma to separate answers as needed.)
Determine the behavior of the function at each zero. Select the correct choice below and, if necessary, fill in the answer boxes within your choice.
A.
The graph touches the x-axis and turns around at all zeros.
B.
The graph crosses the x-axis at xequals
1. The graph touches the x-axis and turns around at xequals
2.
(Type integers or decimals. Simplify your answers. Use a comma to separate answers as needed.)
C.
The graph crosses the x-axis at all zeros.
Answers
GPT-4o mini
To find the zeros of the polynomial function \( f(x) = x^3 - 22x^2 + 121x \), we can factor the polynomial.
1. **Factor out the greatest common factor (GCF)**:
We can factor out \( x \):
\[
f(x) = x(x^2 - 22x + 121)
\]
2. **Factor the quadratic** \( x^2 - 22x + 121 \):
We can try to factor this further:
\[
x^2 - 22x + 121 = (x - 11)^2
\]
Therefore, we can rewrite the polynomial as:
\[
f(x) = x(x - 11)^2
\]
3. **Find the zeros**:
Setting \( f(x) = 0 \):
\[
x(x - 11)^2 = 0
\]
This gives us:
\[
x = 0 \quad \text{and} \quad (x - 11)^2 = 0 \Rightarrow x = 11
\]
4. The zeros are \( x = 0 \) and \( x = 11 \).
5. **Determine the multiplicity of each zero**:
- The zero \( x = 0 \) has a multiplicity of 1.
- The zero \( x = 11 \) has a multiplicity of 2.
6. **Determine the behavior of the graph at each zero**:
- For \( x = 0 \) (multiplicity 1): Since the multiplicity is odd, the graph crosses the \( x \)-axis.
- For \( x = 11 \) (multiplicity 2): Since the multiplicity is even, the graph touches the \( x \)-axis and turns around.
### Final Answers:
- The zeros are:
\[
0, 11
\]
- The behavior of the function at each zero:
- A. The graph touches the \( x \)-axis and turns around at all zeros.
- Incorrect.
- B. The graph crosses the \( x \)-axis at \( x = 0 \) and touches the \( x \)-axis and turns around at \( x = 11 \).
- Select this option:
- 1. The graph crosses the \( x \)-axis at \( x = 0 \).
- 2. The graph touches the \( x \)-axis and turns around at \( x = 11 \).
- This will be the final answer: Choose option B with the appropriate input.
- C. The graph crosses the \( x \)-axis at all zeros.
- Incorrect.
### Summary:
- **Zeros:** \( 0, 11 \)
- **Behavior:** Crosses at \( 0 \), touches and turns at \( 11 \).
1. **Factor out the greatest common factor (GCF)**:
We can factor out \( x \):
\[
f(x) = x(x^2 - 22x + 121)
\]
2. **Factor the quadratic** \( x^2 - 22x + 121 \):
We can try to factor this further:
\[
x^2 - 22x + 121 = (x - 11)^2
\]
Therefore, we can rewrite the polynomial as:
\[
f(x) = x(x - 11)^2
\]
3. **Find the zeros**:
Setting \( f(x) = 0 \):
\[
x(x - 11)^2 = 0
\]
This gives us:
\[
x = 0 \quad \text{and} \quad (x - 11)^2 = 0 \Rightarrow x = 11
\]
4. The zeros are \( x = 0 \) and \( x = 11 \).
5. **Determine the multiplicity of each zero**:
- The zero \( x = 0 \) has a multiplicity of 1.
- The zero \( x = 11 \) has a multiplicity of 2.
6. **Determine the behavior of the graph at each zero**:
- For \( x = 0 \) (multiplicity 1): Since the multiplicity is odd, the graph crosses the \( x \)-axis.
- For \( x = 11 \) (multiplicity 2): Since the multiplicity is even, the graph touches the \( x \)-axis and turns around.
### Final Answers:
- The zeros are:
\[
0, 11
\]
- The behavior of the function at each zero:
- A. The graph touches the \( x \)-axis and turns around at all zeros.
- Incorrect.
- B. The graph crosses the \( x \)-axis at \( x = 0 \) and touches the \( x \)-axis and turns around at \( x = 11 \).
- Select this option:
- 1. The graph crosses the \( x \)-axis at \( x = 0 \).
- 2. The graph touches the \( x \)-axis and turns around at \( x = 11 \).
- This will be the final answer: Choose option B with the appropriate input.
- C. The graph crosses the \( x \)-axis at all zeros.
- Incorrect.
### Summary:
- **Zeros:** \( 0, 11 \)
- **Behavior:** Crosses at \( 0 \), touches and turns at \( 11 \).