Relative Extrema of Polynomials Quick Check

Question

5 of 55 of 5 Items

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

eight at most
eight at most

three at most
three at most

exactly eight
exactly eight

nine at most

Answer 1

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

1. First, let's determine the degree of the polynomial:

   • 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.
   • Therefore, the total degree of the polynomial is \( 3 + 2 + 4 = 9 \).

2. The maximum number of turning points of a polynomial is one less than its degree. So, for a polynomial of degree 9, the maximum number of turning points is \( 9 - 1 = 8 \).

Considering this information, Rogelio's sketch of \( g(x) \) should include:

eight at most.