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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
negative 1.5

2
2

1.5
1.5

1

Answer 1

To determine the possible multiplicity of the factor \( (2x + 3) \) in the polynomial \( p(x) \), we need to consider the properties of polynomial functions and their roots.

1. Multiplicity and Crossings: The multiplicity of a root affects how the graph behaves at that root. If the multiplicity is odd, the graph will cross the x-axis at that root, and if it is even, the graph will touch the x-axis without crossing it.

2. Given Information About the Graph:

   • The graph \( p(x) \) has turning points and inflection points, meaning it changes direction multiple times.
   • Since \( p(x) \) has turning points, the factors corresponding to the roots are likely to have odd multiplicities at points where the polynomial crosses the x-axis.

3. Analysis of the Options:

   • −1.5: This is not a valid multiplicity since multiplicity must be a non-negative integer.
   • 1.5: This is also not a valid multiplicity because it is not an integer.
   • 2: This is valid and indicates that the graph would touch the x-axis at this root, not crossing it.
   • 1: This is valid and indicates that the graph would cross the x-axis at this root.

Since the behavior of the polynomial \( p(x) \) appears to cross the axis rather than only touch it (specifically indicated by the inflection points), it is more likely for \( (2x + 3) \) to have a multiplicity of 1.

Thus, the answer would be 1.