Bot GPT 4-o mini To determine whether the factor \( (2x + 1) \) has an even or odd multiplicity in the polynomial \( w(x) \), we need to analyze the behavior of the graph of \( w(x) \) at the point where it crosses or touches the x-axis corresponding to the root from the factor \( (2x + 1) = 0 \).
The root of \( (2x + 1) \) occurs when \( x = -\frac{1}{2} \).
- If the graph of \( w(x) \) crosses the x-axis at \( x = -\frac{1}{2} \), it indicates that the multiplicity is odd.
- If the graph of \( w(x) \) touches the x-axis and turns around at \( x = -\frac{1}{2} \), it indicates that the multiplicity is even.
Without seeing the graph, I can't definitively tell you the multiplicity. However, you can use this guideline to make your determination based on the graph's behavior at \( x = -\frac{1}{2} \).
So, you would reply with "Even" if it touches the axis (even multiplicity) or "Odd" if it crosses it (odd multiplicity).
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