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?

A
at most 2

B
at most 7

C
at most 8

D
at most 10

Answer 1

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

1. **Identify the factors** and their degrees:
- The factor \( (2x + 5) \) is a linear factor (degree 1).
- The factor \( (x - 3)^3 \) is a cubic factor (degree 3).
- The factor \( (8x - 1)^5 \) is a quintic factor (degree 5).

2. **Calculate the total degree of \( m(x) \)**:
- The total degree of the polynomial is the sum of the degrees of each factor:
\[
\text{Total Degree} = 1 + 3 + 5 = 9
\]

3. **Determine the maximum number of turning points**:
- A polynomial of degree \( n \) can have at most \( n - 1 \) turning points. Thus, for this polynomial:
\[
\text{Maximum Turning Points} = 9 - 1 = 8
\]

Therefore, Brandy's sketch should include **at most 8 turning points**.

The correct answer is **C: at most 8**.