Question
Wilma is asked to sketch a graph of m(x)=(7x-1)^7(x+3)^2(5x+2)
. How many turning points should her sketch include?
A
at most 5
B
at most 9
C
at most 7
D
at most 10
Answers
GPT-4o mini
To determine how many turning points the function \( m(x) = (7x-1)^7(x+3)^2(5x+2) \) can have, we first need to analyze the degree of the polynomial.
1. **Identify the degrees of each factor**:
- \( (7x - 1)^7 \) has a degree of 7.
- \( (x + 3)^2 \) has a degree of 2.
- \( (5x + 2) \) has a degree of 1.
2. **Add the degrees together** to find the overall degree of \( m(x) \):
\[
\text{Degree of } m(x) = 7 + 2 + 1 = 10.
\]
3. **Turning points**: A polynomial of degree \( n \) can have at most \( n - 1 \) turning points.
For \( m(x) \) which is of degree 10:
\[
\text{Maximum number of turning points} = 10 - 1 = 9.
\]
Thus, for her sketch, Wilma should include at most **9** turning points.
The correct answer is **B: at most 9**.
1. **Identify the degrees of each factor**:
- \( (7x - 1)^7 \) has a degree of 7.
- \( (x + 3)^2 \) has a degree of 2.
- \( (5x + 2) \) has a degree of 1.
2. **Add the degrees together** to find the overall degree of \( m(x) \):
\[
\text{Degree of } m(x) = 7 + 2 + 1 = 10.
\]
3. **Turning points**: A polynomial of degree \( n \) can have at most \( n - 1 \) turning points.
For \( m(x) \) which is of degree 10:
\[
\text{Maximum number of turning points} = 10 - 1 = 9.
\]
Thus, for her sketch, Wilma should include at most **9** turning points.
The correct answer is **B: at most 9**.