To determine the number of turning points of the function \( m(x) = (2x + 5)(x - 3)^3(8x + 1)^5 \), we need to analyze the polynomial.
The function \( m(x) \) is a polynomial, and the number of turning points of a polynomial of degree \( n \) can be at most \( n - 1 \).
First, we need to find the degree of \( m(x) \):
- The term \( (2x + 5) \) has degree 1.
- The term \( (x - 3)^3 \) has degree 3.
- The term \( (8x + 1)^5 \) has degree 5.
Now, we add the degrees of the individual terms: \[ \text{Degree of } m(x) = 1 + 3 + 5 = 9. \]
Since a polynomial of degree 9 can have at most \( 9 - 1 = 8 \) turning points, Brandy should include 8 turning points in her sketch.
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