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)
o eight at most
o three at most
o exactly eight
o nine at most

Explanation:

Step1: Recall the degree - turning point relationship

The number of turning points of a polynomial function \(y = f(x)\) is at most \(n - 1\), where \(n\) is the degree of the polynomial. First, find the degree of the function \(g(x)=3x^{3}(x - 5)^{2}(5 - x)^{4}\).

Step2: Calculate the degree of the polynomial

Using the rule of exponents for multiplying powers with the same base (\(a^{m}\cdot a^{n}=a^{m + n}\)) and the fact that \((5 - x)^{4}=(x - 5)^{4}\), we have \(g(x)=3x^{3}(x - 5)^{2}(x - 5)^{4}=3x^{3}(x - 5)^{6}\). The degree of the polynomial is the sum of the exponents of the factors, so \(n=3 + 6=9\).

Step3: Determine the number of turning points

Since the number of turning points of a polynomial function is at most \(n - 1\), for \(n = 9\), the number of turning points is at most \(9-1 = 8\).

Answer:

eight at most