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

exactly eight
exactly eight

three at most
three at most

eight at most
eight at most

nine at most

Answer 1

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

1. **Degree of the polynomial**:
The degree of a polynomial is found by adding the degrees of its factors. Here's the breakdown for the given function:
- \( 3x^3 \) contributes a degree of 3.
- \( (x-5)^2 \) contributes a degree of 2.
- \( (5-x)^4 \) contributes a degree of 4 (since \( (5-x) \) is equivalent to \( -(x-5) \), it contributes the same degree).

Therefore, the total degree of \( g(x) \) is:
\[
3 + 2 + 4 = 9
\]

2. **Maximum number of turning points**:
A polynomial of degree \( n \) can have at most \( n - 1 \) turning points. For our polynomial of degree 9, this means:
\[
9 - 1 = 8 \text{ turning points at most.}
\]

Thus, the correct answer is:
**eight at most**.