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)

Question

Responses

nine at most
nine at most

eight at most
eight at most

three at most
three at most

exactly eight

Answer 1

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

1. Identify the degree of each factor:

   • The factor \( 3x^3 \) contributes a degree of 3.
   • The factor \( (x-5)^2 \) contributes a degree of 2.
   • The factor \( (5-x)^4 \) also contributes a degree of 4.

2. Calculate the total degree of the polynomial: \[ \text{Total degree} = 3 + 2 + 4 = 9 \]

3. Determine the maximum number of turning points: The maximum number of turning points of a polynomial is given by the degree minus one. \[ \text{Maximum turning points} = 9 - 1 = 8 \]

Thus, the sketch of the graph should include at most eight turning points.

The correct response is: eight at most.