Question

dentify the following given
f
(
x
)
=
(
x
+
5
)
2
(
2
x
−
3
)
5
f(x)=(x+5)
2
(2x−3)
5




Zeros & Multiplicity

For each zero state number of its multiplicity.

​
x
=
−
5
x=−5 multiplicity is


​
x
=
3
2
x=
2
3
​
multiplicity is




Zeros & Effect

For each zero state its effect.

​
x
=
−
5
x=−5 effect is


​
x
=
3
2
x=
2
3
​
effect is


​

Answer 1

To identify the zeros and their multiplicities for the function \( f(x) = (x + 5)^2 (2x - 3)^5 \), we can follow these steps:

### Zeros and Their Multiplicities

1. **Finding Zeros:**
- For the first factor \( (x + 5)^2 = 0 \):
\[
x + 5 = 0 \implies x = -5
\]
The multiplicity of the zero at \( x = -5 \) is \( 2 \) (since it is squared).

- For the second factor \( (2x - 3)^5 = 0 \):
\[
2x - 3 = 0 \implies 2x = 3 \implies x = \frac{3}{2}
\]
The multiplicity of the zero at \( x = \frac{3}{2} \) is \( 5 \) (since it is raised to the fifth power).

### Summary of Zeros and Their Multiplicities:

- **\( x = -5 \)**
- Multiplicity: \( 2 \)

- **\( x = \frac{3}{2} \)**
- Multiplicity: \( 5 \)

### Effect of Each Zero

The effect of a zero can be understood in terms of how it influences the graph of the function at that point:

1. **For the zero \( x = -5 \)**:
- Since the multiplicity is **even** (2), the graph touches the x-axis but does not cross it. Thus:
- **Effect:** The graph **touches** the x-axis at \( x = -5 \).

2. **For the zero \( x = \frac{3}{2} \)**:
- Since the multiplicity is **odd** (5), the graph crosses the x-axis at this point. Thus:
- **Effect:** The graph **crosses** the x-axis at \( x = \frac{3}{2} \).

### Final Summary:

- **Zeros & Multiplicities:**
- \( x = -5 \) — Multiplicity: **2**
- \( x = \frac{3}{2} \) — Multiplicity: **5**

- **Zeros & Effects:**
- \( x = -5 \) — Effect: **Touches** the x-axis.
- \( x = \frac{3}{2} \) — Effect: **Crosses** the x-axis.