Brandy is asked to sketch a graph of m(x)=(2x+5)(x-3)^3(8x+1)^5

Question

. How many turning points should her sketch include?

A
at most 2

B
at most 7

C
at most 8

D
at most 10

Answer 1

To determine the number of turning points for the function \( m(x) = (2x + 5)(x - 3)^3(8x + 1)^5 \), we first need to analyze the factors in the polynomial.

1. Identify the degree of the polynomial: The degree of a polynomial gives an upper bound on the maximum number of turning points, which is at most \( n - 1 \), where \( n \) is the degree of the polynomial.

   • The term \( (2x + 5) \) contributes a degree of 1.
   • The term \( (x - 3)^3 \) contributes a degree of 3.
   • The term \( (8x + 1)^5 \) contributes a degree of 5.

   Thus, the total degree of \( m(x) \) is: \[ 1 + 3 + 5 = 9. \]

2. Turning points: The number of turning points of a polynomial is at most \( n - 1 \): \[ \text{Maximum turning points} = 9 - 1 = 8. \]

So, Brandy should include at most 8 turning points in her sketch.

Therefore, the correct answer is C: at most 8.