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 maximum number of turning points for the function \( g(x) = 3x^3(x - 5)^2(5 - x)^4 \), we can use the fact that for a polynomial of degree \( n \), the maximum number of turning points is \( n - 1 \).

First, we need to find the degree of the polynomial \( g(x) \):

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

Adding these contributions together gives us the total degree: \[ 3 + 2 + 4 = 9 \]

Since the degree of the polynomial \( g(x) \) is 9, the maximum number of turning points is: \[ 9 - 1 = 8 \]

Thus, Rogelio's sketch should include exactly eight turning points.

Answer 2

Are you sure

Answer 3

Let's verify the calculation step by step to ensure accuracy regarding the polynomial \( g(x) = 3x^3(x - 5)^2(5 - x)^4 \).

1. Identify the degrees of each factor:

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

2. Total degree: The total degree of \( g(x) \) is the sum of the degrees of its factors: \[ 3 + 2 + 4 = 9 \]

3. Maximum number of turning points: The maximum number of turning points for a polynomial of degree \( n \) is given by \( n - 1 \). Since the degree of \( g(x) \) is 9, we calculate the maximum number of turning points as follows: \[ 9 - 1 = 8 \]

4. Conclusion: So, the function \( g(x) = 3x^3(x - 5)^2(5 - x)^4 \) can indeed have a maximum of 8 turning points.

Given this confirmation, the correct answer is indeed that Rogelio's sketch should include exactly eight turning points.