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?(1 point)
Responses

eight at most
eight at most

10 at most
10 at most

two at most
two at most

seven at most
seven at most

Answer 1

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

The function is a product of several factors, and the degree of the polynomial can be found by adding the degrees of each factor:

1. \( (2x + 5) \) is a linear factor of degree 1.
2. \( (x - 3)^3 \) is a polynomial factor of degree 3.
3. \( (8x + 1)^5 \) is a polynomial factor of degree 5.

Now, we calculate the total degree:

• The degree contributed by \( (2x + 5) \) is 1.
• The degree contributed by \( (x - 3)^3 \) is 3.
• The degree contributed by \( (8x + 1)^5 \) is 5.

Adding these together, we have: \[ 1 + 3 + 5 = 9 \]

For a polynomial of degree \( n \), the maximum number of turning points is \( n - 1 \). Therefore, for a polynomial of degree 9, the maximum number of turning points is: \[ 9 - 1 = 8 \]

Thus, Brandy should include eight at most turning points in her sketch.

So the correct response is: eight at most.