To determine the maximum number of turning points in the function \( m(x) = (2x+5)(x-3)^3(8x-1)^5 \), we first need to understand the degree of the polynomial.
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Identify the factors and their degrees:
- The factor \( (2x + 5) \) is a linear factor (degree 1).
- The factor \( (x - 3)^3 \) is a cubic factor (degree 3).
- The factor \( (8x - 1)^5 \) is a quintic factor (degree 5).
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Calculate the total degree of \( m(x) \):
- The total degree of the polynomial is the sum of the degrees of each factor: \[ \text{Total Degree} = 1 + 3 + 5 = 9 \]
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Determine the maximum number of turning points:
- A polynomial of degree \( n \) can have at most \( n - 1 \) turning points. Thus, for this polynomial: \[ \text{Maximum Turning Points} = 9 - 1 = 8 \]
Therefore, Brandy's sketch should include at most 8 turning points.
The correct answer is C: at most 8.
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