Question
Consider the graph of the polynomial function p(x). Given that is a factor of p(x),
which of the following could be the multiplicity of ?
(1 point)
(2x + 3) ( 2𝑥 + 3 )
(2x + 3) ( 2𝑥 + 3 )
−1.5 −1.5
2
1
1.5
which of the following could be the multiplicity of ?
(1 point)
(2x + 3) ( 2𝑥 + 3 )
(2x + 3) ( 2𝑥 + 3 )
−1.5 −1.5
2
1
1.5
Answers
Answered by
GPT-4o mini
To determine the possible multiplicity of the linear factor \(2x + 3\) in the polynomial function \(p(x)\), we consider the conditions under which a polynomial has a factor.
1. **Multiplicity Explanation**:
- A factor of a polynomial can have a multiplicity of \(1\) (the graph crosses the x-axis at that root) or higher values like \(2\), \(3\), etc. (that indicates the graph touches the x-axis at that root and does not cross it).
- For the given factor \(2x + 3\), the corresponding root is \(x = -\frac{3}{2} = -1.5\).
2. **Possible Multiplicities**:
- If the multiplicity is \(1\), the graph will cross the x-axis at x = -1.5.
- If the multiplicity is \(2\), the graph will touch the x-axis at x = -1.5 and turn back.
- Multiplicities greater than \(2\) also indicate the graph will touch the x-axis and may behave differently based on the even or odd nature.
3. **Considering the Multiple Choice Options**:
- **2**: Possible (the graph could touch the x-axis at -1.5).
- **1**: Possible (the graph could cross the x-axis at -1.5).
- **1.5**: Not valid as multiplicities must be whole numbers.
- **-1.5**: Not valid as multiplicities must be non-negative integers.
Thus, based on the provided choices, the possible multiplicities for the factor \(2x + 3\) in the polynomial function \(p(x)\) could be \(2\) or \(1\). The only values that count as valid options from your list are:
- **1**
- **2**
If you must choose only one from your provided options, select either **1** or **2**, as both are valid multiplicities. Let me know if you need further clarification!
1. **Multiplicity Explanation**:
- A factor of a polynomial can have a multiplicity of \(1\) (the graph crosses the x-axis at that root) or higher values like \(2\), \(3\), etc. (that indicates the graph touches the x-axis at that root and does not cross it).
- For the given factor \(2x + 3\), the corresponding root is \(x = -\frac{3}{2} = -1.5\).
2. **Possible Multiplicities**:
- If the multiplicity is \(1\), the graph will cross the x-axis at x = -1.5.
- If the multiplicity is \(2\), the graph will touch the x-axis at x = -1.5 and turn back.
- Multiplicities greater than \(2\) also indicate the graph will touch the x-axis and may behave differently based on the even or odd nature.
3. **Considering the Multiple Choice Options**:
- **2**: Possible (the graph could touch the x-axis at -1.5).
- **1**: Possible (the graph could cross the x-axis at -1.5).
- **1.5**: Not valid as multiplicities must be whole numbers.
- **-1.5**: Not valid as multiplicities must be non-negative integers.
Thus, based on the provided choices, the possible multiplicities for the factor \(2x + 3\) in the polynomial function \(p(x)\) could be \(2\) or \(1\). The only values that count as valid options from your list are:
- **1**
- **2**
If you must choose only one from your provided options, select either **1** or **2**, as both are valid multiplicities. Let me know if you need further clarification!