Question

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

nine at most
nine at most

eight at most
eight at most

three at most
three at most

exactly eight
exactly eight

Answer 1

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

1. Identify the degree of the polynomial:

   • The degree of \( g(x) \) is calculated by adding the exponents of each term:
     • From \( 3x^3 \), the degree contributes 3.
     • From \( (x - 5)^2 \), the degree contributes 2.
     • From \( (5 - x)^4 \), the degree contributes 4.
   • Therefore, the total degree of \( g(x) \) is \( 3 + 2 + 4 = 9 \).

2. Determine the maximum number of turning points:

   • A polynomial function can have at most \( n - 1 \) turning points, where \( n \) is the degree of the polynomial.
   • Here, with a degree of 9, the maximum number of turning points is \( 9 - 1 = 8 \).

Thus, the correct response to how many turning points Rogelio should include in his sketch is eight at most.