Toplic realize. Topiew (LMS graded)
Part 1 of 2
Find the zeros of the function, and then describe the behavior of the graph at each zero.

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Dec 27,2024

Question

Toplic realize. Topiew (LMS graded) Part 1 of 2 Find the zeros of the function, and then describe the behavior of the graph at each zero. $$ f(x)=x^{3}+4 x^{2}+4 x $$

UpStudy AI · Accepted Answer

To find the zeros of the function $$ f(x) = x^{3} + 4x^{2} + 4x $$ and describe the behavior of the graph at each zero, follow these steps:

### 1. **Factor the Function**

First, factor the given cubic function:

$$
f(x) = x^{3} + 4x^{2} + 4x
$$

Factor out the greatest common factor (which is $$ x $$):

$$
f(x) = x(x^{2} + 4x + 4)
$$

Next, factor the quadratic expression $$ x^{2} + 4x + 4 $$:

$$
x^{2} + 4x + 4 = (x + 2)^2
$$

So, the fully factored form of the function is:

$$
f(x) = x(x + 2)^2
$$

### 2. **Find the Zeros**

Set each factor equal to zero to find the zeros:

- **First Zero:**

$$
  x = 0
  $$

- **Second Zero:**

$$
  x + 2 = 0 \quad \Rightarrow \quad x = -2
  $$

Thus, the zeros of the function are:

$$
x = 0 \quad \text{and} \quad x = -2
$$

### 3. **Determine the Multiplicity of Each Zero**

- **Zero at $$ x = 0 $$:** This zero comes from the factor $$ x $$ and has a **multiplicity of 1**.
  
- **Zero at $$ x = -2 $$:** This zero comes from the factor $$ (x + 2)^2 $$ and has a **multiplicity of 2**.

### 4. **Describe the Behavior at Each Zero**

- **At $$ x = 0 $$ (Multiplicity 1):**
  
  - **Multiplicity is odd (1).**
  - **Behavior:** The graph **crosses** the x-axis at this point.

- **At $$ x = -2 $$ (Multiplicity 2):**
  
  - **Multiplicity is even (2).**
  - **Behavior:** The graph **touches** the x-axis and **turns around** at this point without crossing.

### 5. **Summary**

- **Zeros:**
  - $$ x = 0 $$ (Multiplicity 1)
  - $$ x = -2 $$ (Multiplicity 2)

- **Graph Behavior:**
  - **At $$ x = 0 $$:** The graph crosses the x-axis.
  - **At $$ x = -2 $$:** The graph touches the x-axis and turns around.

### 6. **Graphical Illustration**

Here's a rough sketch to visualize the behavior:

```
       |
     f(x)
       |
    *--•--*--•
       |
      -2   0
```

- The graph **crosses** the x-axis at $$ x = 0 $$.
- The graph **touches** and **turns around** at $$ x = -2 $$.

This analysis fully describes the zeros and the corresponding behavior of the graph for the given function.
The zeros of the function $$ f(x) = x^{3} + 4x^{2} + 4x $$ are $$ x = 0 $$ and $$ x = -2 $$. At $$ x = 0 $$, the graph crosses the x-axis, and at $$ x = -2 $$, it touches the x-axis and turns around without crossing.

Toplic realize. Topiew (LMS graded)
Part 1 of 2
Find the zeros of the function, and then describe the behavior of the graph at each zero.

Solución de inteligencia artificial de Upstudy

Responder

Solución

1. Factor the Function

2. Find the Zeros

3. Determine the Multiplicity of Each Zero

4. Describe the Behavior at Each Zero

5. Summary

6. Graphical Illustration

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