Question
Brandy is asked to sketch a graph of m(x) = (2x + 5)(x − 3)3(8x + 1)5
(1 point)
Responses
two at most
two at most
eight at most
eight at most
10 at most
10 at most
seven at most
(1 point)
Responses
two at most
two at most
eight at most
eight at most
10 at most
10 at most
seven at most
Answers
GPT-4o mini
To determine the maximum number of times the graph of the function \( m(x) = (2x + 5)(x - 3)^3(8x + 1)^5 \) can intersect the x-axis, we need to analyze the roots and their multiplicities:
1. **Root from \( (2x + 5) \)**:
- Set \( 2x + 5 = 0 \) which gives \( x = -\frac{5}{2} \). This root has a multiplicity of 1.
2. **Root from \( (x - 3)^3 \)**:
- Set \( x - 3 = 0 \) which gives \( x = 3 \). This root has a multiplicity of 3.
3. **Root from \( (8x + 1)^5 \)**:
- Set \( 8x + 1 = 0 \) which gives \( x = -\frac{1}{8} \). This root has a multiplicity of 5.
Now, to find the total number of roots (which corresponds to the maximum number of x-axis intersections), we add the multiplicities:
- From \( (2x + 5) \): **1**
- From \( (x - 3)^3 \): **3**
- From \( (8x + 1)^5 \): **5**
Adding these up:
\[
1 + 3 + 5 = 9
\]
Therefore, the maximum number of times the graph of \( m(x) \) can intersect the x-axis is **9**.
Since the closest answer in your given choices is "10 at most", if "10 at most" means "can intersect at most 10 times", then you would select that. However, based on the function provided, it actually intersects the x-axis **9 times**.
Thus, in this context, your answer would align with the response "10 at most".
1. **Root from \( (2x + 5) \)**:
- Set \( 2x + 5 = 0 \) which gives \( x = -\frac{5}{2} \). This root has a multiplicity of 1.
2. **Root from \( (x - 3)^3 \)**:
- Set \( x - 3 = 0 \) which gives \( x = 3 \). This root has a multiplicity of 3.
3. **Root from \( (8x + 1)^5 \)**:
- Set \( 8x + 1 = 0 \) which gives \( x = -\frac{1}{8} \). This root has a multiplicity of 5.
Now, to find the total number of roots (which corresponds to the maximum number of x-axis intersections), we add the multiplicities:
- From \( (2x + 5) \): **1**
- From \( (x - 3)^3 \): **3**
- From \( (8x + 1)^5 \): **5**
Adding these up:
\[
1 + 3 + 5 = 9
\]
Therefore, the maximum number of times the graph of \( m(x) \) can intersect the x-axis is **9**.
Since the closest answer in your given choices is "10 at most", if "10 at most" means "can intersect at most 10 times", then you would select that. However, based on the function provided, it actually intersects the x-axis **9 times**.
Thus, in this context, your answer would align with the response "10 at most".