Question

Brandy is asked to sketch a graph of m(x)=(2x+5)(x−3)3(8x+1)5
. How many turning points should her sketch include?(1 point)
Responses

two at most
two at most

seven at most
seven at most

eight at most
eight at most

10 at most

Answer 1

To determine the maximum number of turning points of the function \( m(x) = (2x + 5)(x - 3)^3(8x + 1)^5 \), we first need to consider the degree of the polynomial.

1. **Identify the degree of each factor:**
- \( (2x + 5) \) is a polynomial of degree 1.
- \( (x - 3)^3 \) is a polynomial of degree 3.
- \( (8x + 1)^5 \) is a polynomial of degree 5.

2. **Calculate the total degree:**
The total degree of \( m(x) \) is the sum of the degrees of its factors:
\[
1 + 3 + 5 = 9.
\]

3. **Determine the maximum number of turning points:**
The maximum number of turning points of a polynomial function is given by \( \text{degree} - 1 \). Therefore, for this polynomial:
\[
9 - 1 = 8.
\]

Thus, Brandy's sketch of the graph should include **eight turning points at most**.

Therefore, the answer is:
**eight at most**.