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

Eight at most
Eight at most

Seven at most
Seven at most

Ten at most

Answer 1

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

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

2. Calculate the total degree of the polynomial \( m(x) \):
\[
\text{Total degree} = 1 + 3 + 5 = 9
\]

3. The number of turning points of a polynomial is at most equal to its degree minus 1. Therefore, for our polynomial of degree 9, the maximum number of turning points is:
\[
\text{Maximum turning points} = 9 - 1 = 8
\]

Thus, Brandy should include **eight at most** turning points in her sketch.

The correct answer is: **Eight at most**.