The statement you provided includes some incorrect interpretations regarding the relationship between the multiplicity of a factor and its behavior on the graph.
If \((x - 3)\) is a factor of \(p(x)\), we first need to understand how the multiplicity of the root (zero) at \(x = 3\) affects the graph of \(p(x)\):
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Odd Multiplicity: If the multiplicity of \((x - 3)\) is odd, the graph will cross the x-axis at \(x = 3\). This means that as \(x\) approaches \(3\), the sign of \(p(x)\) changes as it passes through the point (3, 0).
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Even Multiplicity: If the multiplicity of \((x - 3)\) is even, the graph will touch the x-axis at \(x = 3\) but will not cross it. Instead, the graph bounces back from the x-axis, indicating that the sign of \(p(x)\) does not change as it passes through \(x = 3\).
Based on this understanding, here's the correct answer:
- The multiplicity of \((x - 3)\) is even if the graph touches the x-axis at \( (3, 0) \) and then turns around.
- The multiplicity of \((x - 3)\) is odd if the graph crosses the x-axis at \( (3, 0) \).
In the context of your statements, the correct choice related to the behavior of the graph is:
The multiplicity of (x-3) is even because the graph of p(x) touches the x-axis at (3,0) and then turns around.
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