Question

Rogelio is asked to sketch a graph of g(x)=3x3(x−5)2(5−x)4. How many turning points should his sketch include?(1 point)
Responses

eight at most
eight at most

nine at most
nine at most

exactly eight
exactly eight

three at most

Answer 1

To determine the number of turning points for the function \( g(x) = 3x^3(x - 5)^2(5 - x)^4 \), we first need to identify the degree of this polynomial.

1. **Identify the degree**:
- The term \( x^3 \) contributes a degree of 3.
- The term \( (x - 5)^2 \) contributes a degree of 2.
- The term \( (5 - x)^4 \) is equivalent to \( (-1)^4(x - 5)^4 \), which contributes a degree of 4.

Adding these contributions together:
\[
\text{Degree} = 3 + 2 + 4 = 9
\]

The total degree of the polynomial \( g(x) \) is 9.

2. **Determine the maximum number of turning points**:
The maximum number of turning points for a polynomial is given by \( \text{Degree} - 1 \). Thus for \( g(x) \):
\[
\text{Maximum turning points} = 9 - 1 = 8
\]

Therefore, Rogelio should include **eight turning points** in his sketch of the graph.

The correct response is: **eight at most**.