Question
Brandy is asked to sketch a graph of m(x)=(2x+5)(x−3)3(8x+1)5 . How many turning points should her sketch include?(1 point)Responses10 at most10 at mosteight at mosteight at mosttwo at mosttwo at mostseven at most
Answers
GPT-4o mini
To determine the maximum number of turning points of the function \( m(x) = (2x + 5)(x - 3)^3(8x + 1)^5 \), you can use the following approach:
1. **Identify the degree of the polynomial**: The degree of a polynomial gives a hint about the potential number of turning points. The number of turning points for a polynomial of degree \( n \) can be at most \( n - 1 \).
2. **Calculate the degree of the polynomial**:
- The term \( (2x + 5) \) is of degree 1.
- The term \( (x - 3)^3 \) is of degree 3.
- The term \( (8x + 1)^5 \) is of degree 5.
So, the total degree of \( m(x) \) is:
\[
1 + 3 + 5 = 9
\]
3. **Determine the maximum number of turning points**: Since the polynomial is of degree 9, the maximum number of turning points is:
\[
9 - 1 = 8
\]
Thus, the sketch of the graph of \( m(x) \) should include **eight at most turning points**.
Therefore, the correct response is **eight at most**.
1. **Identify the degree of the polynomial**: The degree of a polynomial gives a hint about the potential number of turning points. The number of turning points for a polynomial of degree \( n \) can be at most \( n - 1 \).
2. **Calculate the degree of the polynomial**:
- The term \( (2x + 5) \) is of degree 1.
- The term \( (x - 3)^3 \) is of degree 3.
- The term \( (8x + 1)^5 \) is of degree 5.
So, the total degree of \( m(x) \) is:
\[
1 + 3 + 5 = 9
\]
3. **Determine the maximum number of turning points**: Since the polynomial is of degree 9, the maximum number of turning points is:
\[
9 - 1 = 8
\]
Thus, the sketch of the graph of \( m(x) \) should include **eight at most turning points**.
Therefore, the correct response is **eight at most**.