Question

Brandy is asked to sketch a graph of m (x) = (2x + 5) (x-3)^2 (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 number of turning points for the function \( m(x) = (2x + 5)(x - 3)^2(8x + 1)^5 \), we can analyze its factors and their respective degrees.

1. **Identify the factors and their contributions to turning points**:
- \( 2x + 5 \) is a linear factor (degree 1), contributing up to 1 turning point.
- \( (x - 3)^2 \) is a quadratic factor (degree 2), contributing up to 2 turning points (however, this factor will only contribute 1 turning point because it is a double root, meaning the graph will touch but not cross the x-axis).
- \( (8x + 1)^5 \) is a polynomial of degree 5, contributing up to 5 turning points (similarly to the linear factor, this will contribute only 5 turning points as it is an odd degree).

2. **Total degree of the polynomial**:
- The total degree of \( m(x) \) is \( 1 + 2 + 5 = 8 \). A polynomial of degree \( n \) can have at most \( n - 1 \) turning points.

Since \( m(x) \) has a total degree of 8, the maximum number of turning points it can have is \( 8 - 1 = 7 \).

Thus, Brandy's sketch should include:

**B) at most 7 turning points.**