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