Question

rogelio is asked to sketch a graph of (g(x)=3x^{3}(x - 5)^{2}(5 - x)^{4}). how many turning points should his sketch include? (1 point) eight at most exactly eight three at most nine at most

Explanation:

Step1: Determine the degree of the polynomial

The degree of the polynomial \(g(x)=3x^{3}(x - 5)^{2}(5 - x)^{4}\) is found by adding the exponents of the factors. The degree of \(x^{3}\) is 3, the degree of \((x - 5)^{2}\) is 2 and the degree of \((5 - x)^{4}=( - 1)^{4}(x - 5)^{4}\) is 4. So the degree \(n=3 + 2+4=9\).

Step2: Recall the relationship between degree and turning - points

The maximum number of turning points of a polynomial function \(y = f(x)\) of degree \(n\) is \(n - 1\).
Since \(n = 9\), the maximum number of turning points is \(n-1=9 - 1=8\).

Answer:

eight at most