Question

A plotted curve with 4 inflection points is labeled p left parenthesis x right parenthesis. The plot starts in quadrant 3, rises steeply to a turning point in quadrant 2, before dropping and rising twice in quadrant 1.
Consider the graph of the polynomial function p(x). Given that (2x+3) is a factor of p(x), which of the following could be the multiplicity of (2x+3) ?

(1 point)
Responses

1.5
1.5

1
1

2
2

−1.5

Answer 1

The function \( p(x) \) has been described as having four inflection points, and it starts in quadrant 3, rises to a turning point in quadrant 2, and then has a series of rises and drops in quadrant 1. Given that \( (2x + 3) \) is a factor of \( p(x) \), we can use some properties of polynomial functions to determine the possible multiplicity of the factor.

1. **Multiplicity Effects on the Graph**:
- If a factor has an **odd multiplicity**, the function will cross the x-axis at that root (zero).
- If a factor has an **even multiplicity**, the graph will touch the x-axis but not cross it at that root.

2. **Given Factor**:
- The factor \( (2x + 3) \) corresponds to the root \( x = -\frac{3}{2} \).

3. **Analyzing Inflection Points**:
- A polynomial can change concavity at inflection points. The presence of four inflection points suggests a fairly complex behavior of the polynomial.
- The root \( x = -\frac{3}{2} \) and how the graph behaves near this root will depend heavily on the multiplicity of the factor \( (2x + 3) \).

Based on the typical behavior of polynomials:
- If \( (2x + 3) \) has a multiplicity of 1, it will cross the x-axis.
- If it has a multiplicity of 2, it will touch the x-axis (even multiplicity).
- Multiplicities that are not whole numbers (like 1.5 or 2.5) are not possible for a polynomial function’s roots.
- A negative multiplicity \(-1.5\) is also not applicable.

4. **Valid Options**:
From the options given:
- **1**: Could be valid as it represents a simple crossing of the x-axis.
- **2**: Also valid as it would imply the polynomial touches the axis without crossing.

Therefore, the feasible multiplicities for \( (2x + 3) \) in relation to \( p(x) \) being a polynomial with the described properties could only logically be \( 1 \) or \( 2 \).

Thus, the appropriate answer would be:

**1 or 2**.