To determine how many turning points the function \( m(x) = (2x + 5)(x - 3)^3(8x + 1)^5 \) should have, we need to analyze the function's behavior.
The number of turning points can be estimated based on the degree of the polynomial. The degree of the polynomial \( m(x) \) is the sum of the degrees of its factors:
- The factor \( (2x + 5) \) is a linear polynomial of degree 1.
- The factor \( (x - 3)^3 \) is a polynomial of degree 3.
- The factor \( (8x + 1)^5 \) is a polynomial of degree 5.
Now we can calculate the total degree of \( m(x) \):
\[ \text{Degree}(m(x)) = 1 + 3 + 5 = 9 \]
A polynomial of degree \( n \) can have at most \( n - 1 \) turning points. Therefore, for \( m(x) \):
\[ \text{Maximum Turning Points} = 9 - 1 = 8 \]
Thus, Brandy's sketch of \( m(x) \) should include up to 8 turning points. However, this is an upper limit; the actual number of turning points could be less depending on the specific shape and behavior of the polynomial.
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