To determine whether the multiplicity of the factor \( (x - 3) \) in the polynomial \( p(x) \) is even or odd, we need to consider the implications of \( (x - 3) \) being a factor of \( p(x) \).
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Multiplicity: The multiplicity of a root refers to how many times that root appears as a factor in a polynomial. If \( (x - 3) \) is a factor of \( p(x) \), then \( x = 3 \) is a root of the polynomial.
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Graph Behavior: The behavior of the graph of a polynomial at its roots depends on the multiplicity of the root:
- If the multiplicity is odd, the graph will cross the x-axis at that root.
- If the multiplicity is even, the graph will touch the x-axis at that root but will not cross it.
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Conclusion: Without additional information about the behavior of the graph of \( p(x) \) at \( x = 3 \), we cannot definitively determine whether the multiplicity is even or odd.
- If the graph crosses the x-axis at \( x = 3 \), then the multiplicity of \( (x - 3) \) is odd.
- If the graph touches the x-axis at \( x = 3 \) but does not cross it, then the multiplicity is even.
If you have more details about the behavior of the graph at \( x = 3 \), it would help in identifying the multiplicity.
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