To determine how many turning points the function \( m(x) = (7x-1)^7(x+3)^2(5x+2) \) can have, we first need to analyze the degree of the polynomial.
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Identify the degrees of each factor:
- \( (7x - 1)^7 \) has a degree of 7.
- \( (x + 3)^2 \) has a degree of 2.
- \( (5x + 2) \) has a degree of 1.
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Add the degrees together to find the overall degree of \( m(x) \): \[ \text{Degree of } m(x) = 7 + 2 + 1 = 10. \]
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Turning points: A polynomial of degree \( n \) can have at most \( n - 1 \) turning points.
For \( m(x) \) which is of degree 10: \[ \text{Maximum number of turning points} = 10 - 1 = 9. \]
Thus, for her sketch, Wilma should include at most 9 turning points.
The correct answer is B: at most 9.