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

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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.

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