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

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.