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

seven at most
seven at most

two at most
two at most

10 at most
10 at most

eight at most

Answer 1

To determine the maximum number of turning points for the polynomial function \( m(x) = (2x + 5)(x - 3)^3(8x + 1)^5 \), we can use the following reasoning:

1. **Identify the degree of the polynomial**: The degree of a polynomial is the sum of the exponents of the factors.
- The factor \( (2x + 5) \) contributes a degree of 1.
- The factor \( (x - 3)^3 \) contributes a degree of 3.
- The factor \( (8x + 1)^5 \) contributes a degree of 5.

Therefore, the total degree of the polynomial \( m(x) \) is:
\[
1 + 3 + 5 = 9
\]

2. **Calculate the maximum number of turning points**: A polynomial of degree \( n \) can have at most \( n - 1 \) turning points.

Given that the degree of \( m(x) \) is 9, the maximum number of turning points is:
\[
9 - 1 = 8
\]

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

So, the correct response is:
**eight at most**.