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Bot GPT 4-o mini To determine the correct factored form of the polynomial \( m(x) \) based on the described graph characteristics, we need to consider the following points:
1. **Number of Turning Points**: The graph has 2 turning points. A polynomial can have at most \( n - 1 \) turning points, where \( n \) is the degree of the polynomial. Therefore, \( m(x) \) must be at least of degree 3.
2. **End Behavior**: The graph starts in quadrant 3 (meaning it approaches negative infinity as \( x \) approaches negative infinity) and ends in quadrant 1 (it approaches positive infinity as \( x \) approaches positive infinity). This suggests that \( m(x) \) is a polynomial with an odd degree (which confirms it is likely degree 3, considering it has only 2 turning points).
3. **Roots (Zeros)**: The roots of \( m(x) \) can be inferred based on where the curve crosses the x-axis. If a polynomial has an odd degree and starts in quadrant 3 and ends in quadrant 1, it must have an odd number of real roots. The options provided include combinations of up to 4 linear factors, indicating possible roots.
Considering all these criteria, let's evaluate the options:
1. **Option 1: \( m(x) = (x+1)(x-3)(x-5)(x-15) \)**
- This is a degree 4 polynomial (4 factors), which can have up to 3 turning points. This does not match the requirement for only 2 turning points.
2. **Option 2: \( m(x) = (x+1)(x-3)(x-5) \)**
- This is a degree 3 polynomial with 3 roots. It can have up to 2 turning points, which fits the condition described. Additionally, the factors suggest that it has real roots.
3. **Option 3: \( m(x) = (x-1)(x+3)(x+5)(x+15) \)**
- Like option 1, this is a degree 4 polynomial, which also would have too many turning points.
4. **Option 4: \( m(x) = (x-1)(x+3)(x+5) \)**
- This is also a degree 3 polynomial with 3 roots. Again, it can have 2 turning points, but the roots indicate it may not fulfill all end behavior specifications (staring in quadrant 3 and ending in quadrant 1).
**Conclusion:**
The second option, **\( m(x) = (x+1)(x-3)(x-5) \)**, is most fitting because it is a cubic polynomial (degree 3) that allows for 2 turning points.
Thus, the correct answer is:
**m(x) = (x + 1)(x - 3)(x - 5)**.