To determine the multiplicity of the factors \((2x + 1)\) and \((4x - 5)\) for the polynomial \(w(x)\), we can look at the graph and observe how the graph behaves at the x-intercepts corresponding to these factors.
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For the factor \((2x + 1)\):
- Set \(2x + 1 = 0\) to find the x-intercept: \[ 2x = -1 \Rightarrow x = -\frac{1}{2} \]
- If the graph touches the x-axis and turns around at \(x = -\frac{1}{2}\) (i.e., the graph does not cross the axis), then the multiplicity of the factor is even.
- If the graph crosses the x-axis at this point, then the multiplicity is odd.
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For the factor \((4x - 5)\):
- Set \(4x - 5 = 0\) to find the x-intercept: \[ 4x = 5 \Rightarrow x = \frac{5}{4} \]
- Apply a similar observation: if the graph touches and turns at \(x = \frac{5}{4}\), then the multiplicity is even. If it crosses, the multiplicity is odd.
Without seeing the actual graph, I cannot give a definitive answer. However, follow these steps based on the behavior of the graph at the identified x-intercepts corresponding to each factor to determine the multiplicities you need. If you have access to the graph, check these points accordingly:
- For \(x = -\frac{1}{2}\) regarding \((2x + 1)\),
- For \(x = \frac{5}{4}\) regarding \((4x - 5)\).
If you provide more specifics about the graph (e.g., whether it touches or crosses the x-axis at those points), I can give you the final answers.
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