Question
dentify the following given
f
(
x
)
=
(
x
+
7
)
3
(
5
x
−
8
)
4
f(x)=(x+7)
3
(5x−8)
4
Zeros & Multiplicity
For each zero state number of its multiplicity.
x
=
−
7
x=−7 multiplicity is
x
=
8
5
x=
5
8
multiplicity is
Zeros & Effect
For each zero state its effect.
x
=
−
7
x=−7 effect is
x
=
8
5
x=
5
8
effect is
f
(
x
)
=
(
x
+
7
)
3
(
5
x
−
8
)
4
f(x)=(x+7)
3
(5x−8)
4
Zeros & Multiplicity
For each zero state number of its multiplicity.
x
=
−
7
x=−7 multiplicity is
x
=
8
5
x=
5
8
multiplicity is
Zeros & Effect
For each zero state its effect.
x
=
−
7
x=−7 effect is
x
=
8
5
x=
5
8
effect is
Answers
GPT-4o mini
To find the zeros and their multiplicities as well as their effects for the function \( f(x) = (x + 7)^3 (5x - 8)^4 \), let's analyze the function step by step.
### Identifying Zeros and Multiplicities
1. **Zero at \( x = -7 \)**:
- From the factor \( (x + 7)^3 \), it is clear that \( x = -7 \) is a root.
- **Multiplicity**: The exponent of \( (x + 7) \) is 3, so the multiplicity of the zero \( x = -7 \) is **3**.
2. **Zero at \( x = \frac{8}{5} \)**:
- From the factor \( (5x - 8)^4 \), we set \( 5x - 8 = 0 \) which gives us \( 5x = 8 \) or \( x = \frac{8}{5} \).
- **Multiplicity**: The exponent of \( (5x - 8) \) is 4, so the multiplicity of the zero \( x = \frac{8}{5} \) is **4**.
### Zeros and Their Effects
Next, we can determine the effects of the zeros on the function:
1. **Effect at \( x = -7 \)**:
- Since this zero has an **odd multiplicity** (3), the effect is that the graph of the function will **cross the x-axis** at \( x = -7 \).
2. **Effect at \( x = \frac{8}{5} \)**:
- Since this zero has an **even multiplicity** (4), the effect is that the graph of the function will **touch the x-axis** and **bounce** at \( x = \frac{8}{5} \).
### Summary
- **Zeros & Multiplicities**:
- \( x = -7 \), Multiplicity = **3**
- \( x = \frac{8}{5} \), Multiplicity = **4**
- **Zeros & Effects**:
- \( x = -7 \), Effect = **crosses the x-axis**
- \( x = \frac{8}{5} \), Effect = **touches and bounces off the x-axis**
### Identifying Zeros and Multiplicities
1. **Zero at \( x = -7 \)**:
- From the factor \( (x + 7)^3 \), it is clear that \( x = -7 \) is a root.
- **Multiplicity**: The exponent of \( (x + 7) \) is 3, so the multiplicity of the zero \( x = -7 \) is **3**.
2. **Zero at \( x = \frac{8}{5} \)**:
- From the factor \( (5x - 8)^4 \), we set \( 5x - 8 = 0 \) which gives us \( 5x = 8 \) or \( x = \frac{8}{5} \).
- **Multiplicity**: The exponent of \( (5x - 8) \) is 4, so the multiplicity of the zero \( x = \frac{8}{5} \) is **4**.
### Zeros and Their Effects
Next, we can determine the effects of the zeros on the function:
1. **Effect at \( x = -7 \)**:
- Since this zero has an **odd multiplicity** (3), the effect is that the graph of the function will **cross the x-axis** at \( x = -7 \).
2. **Effect at \( x = \frac{8}{5} \)**:
- Since this zero has an **even multiplicity** (4), the effect is that the graph of the function will **touch the x-axis** and **bounce** at \( x = \frac{8}{5} \).
### Summary
- **Zeros & Multiplicities**:
- \( x = -7 \), Multiplicity = **3**
- \( x = \frac{8}{5} \), Multiplicity = **4**
- **Zeros & Effects**:
- \( x = -7 \), Effect = **crosses the x-axis**
- \( x = \frac{8}{5} \), Effect = **touches and bounces off the x-axis**