In the context of algebra and polynomial functions, the "multiplicity" of a factor refers to the number of times that factor appears in a polynomial. Specifically, if a polynomial has a root at \( r \) and can be expressed as \( (x - r)^m \), where \( m \) is the multiplicity, then:
- If \( m \) is even, the graph of the polynomial touches the x-axis at \( r \) but does not cross it.
- If \( m \) is odd, the graph crosses the x-axis at \( r \).
For the factor \( (4x - 5) \), or equivalently \( 4(x - \frac{5}{4}) \), if we were to analyze the multiplicity of this factor, we would typically consider it to have a multiplicity of 1, since it appears only once.
If your question indicates that the answer you're looking for is about how to make the multiplicity of \( (4x - 5) \) even, you would need to express the factor with an even exponent. For example:
\[ (4x - 5)^2 \]
In this case, \( (4x - 5) \) has a multiplicity of 2, which is even.
If you have a different context or specific details about a problem regarding multiplicity, feel free to provide that, and I can assist you further!
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