Question
Relative Extrema of Polynomials Quick Check
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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
Answers
GPT-4o mini
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**.
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**.