Answer :
Certainly! Let's delve into the process of graphing the polynomial function [tex]\( g(x) = -x(x-3)^2(x+2)^2 \)[/tex]. We'll analyze the function step by step, focusing on its key features such as roots, end behavior, and turning points.
### Step 1: Identify the Roots
First, let's find the roots (or zeros) of the polynomial, which occur when [tex]\( g(x) = 0 \)[/tex].
Set [tex]\( g(x) = 0 \)[/tex]:
[tex]\[ -x(x-3)^2(x+2)^2 = 0 \][/tex]
This equation will be zero if any of the factors are zero:
1. [tex]\( -x = 0 \)[/tex] implies [tex]\( x = 0 \)[/tex]
2. [tex]\( (x-3)^2 = 0 \)[/tex] implies [tex]\( x = 3 \)[/tex]
3. [tex]\( (x+2)^2 = 0 \)[/tex] implies [tex]\( x = -2 \)[/tex]
Therefore, the roots are at [tex]\( x = 0 \)[/tex], [tex]\( x = 3 \)[/tex], and [tex]\( x = -2 \)[/tex].
### Step 2: Determine Multiplicity of Roots
The multiplicity of a root affects the graph's behavior at that root.
- [tex]\( x = 0 \)[/tex]: This root has multiplicity 1 (since it appears as [tex]\( x \)[/tex]).
- [tex]\( x = 3 \)[/tex]: This root has multiplicity 2 (since it appears as [tex]\( (x-3)^2 \)[/tex]).
- [tex]\( x = -2 \)[/tex]: This root has multiplicity 2 (since it appears as [tex]\( (x+2)^2 \)[/tex]).
### Step 3: End Behavior of the Polynomial
Consider the leading term to determine the polynomial's end behavior. The highest degree term in [tex]\( g(x) \)[/tex] is obtained by multiplying the leading terms in each factor:
[tex]\[ -x \cdot x^2 \cdot x^2 = -x^5 \][/tex]
As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex]:
[tex]\[ g(x) \to -x^5 \to -\infty \][/tex]
As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex]:
[tex]\[ g(x) \to -x^5 \to \infty \][/tex]
Therefore, the graph will fall to [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] goes to [tex]\( \infty \)[/tex] and rise to [tex]\( \infty \)[/tex] as [tex]\( x \)[/tex] goes to [tex]\( -\infty \)[/tex].
### Step 4: Behavior Near the Roots
- At [tex]\( x = 0 \)[/tex] (multiplicity 1): The graph crosses the x-axis.
- At [tex]\( x = 3 \)[/tex] (multiplicity 2): The graph touches the x-axis and turns around, forming a local minimum or maximum.
- At [tex]\( x = -2 \)[/tex] (multiplicity 2): The graph touches the x-axis and turns around, forming a local minimum or maximum.
### Step 5: Additional Points and Sketch
To better understand the shape of the graph, evaluate the function at additional points, particularly near the roots and around where the graph turns.
#### At [tex]\( x = -3 \)[/tex]:
[tex]\[ g(-3) = -(-3) \cdot (-3-3)^2 \cdot (-3+2)^2 = 3 \cdot 36 \cdot 1 = 108 \][/tex]
#### At [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = -(-1) \cdot (-1-3)^2 \cdot (-1+2)^2 = 1 \cdot 16 \cdot 1 = -16 \][/tex]
#### At [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = -1 \cdot (1-3)^2 \cdot (1+2)^2 = -1 \cdot 4 \cdot 9 = -36 \][/tex]
#### At [tex]\( x = 4 \)[/tex]:
[tex]\[ g(4) = -4 \cdot (4-3)^2 \cdot (4+2)^2 = -4 \cdot 1 \cdot 36 = -144 \][/tex]
### Sketch the Graph
With this information, we can sketch a rough graph:
1. The graph rises to [tex]\( \infty \)[/tex] as [tex]\( x \)[/tex] goes to [tex]\( -\infty \)[/tex].
2. It crosses the x-axis at [tex]\( x = 0 \)[/tex].
3. It touches and turns around at [tex]\( x = -2 \)[/tex] and [tex]\( x = 3 \)[/tex].
4. It has significant points at [tex]\( (-3, 108) \)[/tex], [tex]\( (-1, -16) \)[/tex], [tex]\( (1, -36) \)[/tex], and [tex]\( (4, -144) \)[/tex].
The overall shape should reflect these key features: starting from [tex]\( \infty \)[/tex] on the left, turning around at [tex]\( x = -2 \)[/tex], crossing at [tex]\( x = 0 \)[/tex], turning around again at [tex]\( x = 3 \)[/tex], and finally falling towards [tex]\( -\infty \)[/tex] on the right.
This completes the detailed examination of the function [tex]\( g(x) = -x(x-3)^2(x+2)^2 \)[/tex] and its graph.
### Step 1: Identify the Roots
First, let's find the roots (or zeros) of the polynomial, which occur when [tex]\( g(x) = 0 \)[/tex].
Set [tex]\( g(x) = 0 \)[/tex]:
[tex]\[ -x(x-3)^2(x+2)^2 = 0 \][/tex]
This equation will be zero if any of the factors are zero:
1. [tex]\( -x = 0 \)[/tex] implies [tex]\( x = 0 \)[/tex]
2. [tex]\( (x-3)^2 = 0 \)[/tex] implies [tex]\( x = 3 \)[/tex]
3. [tex]\( (x+2)^2 = 0 \)[/tex] implies [tex]\( x = -2 \)[/tex]
Therefore, the roots are at [tex]\( x = 0 \)[/tex], [tex]\( x = 3 \)[/tex], and [tex]\( x = -2 \)[/tex].
### Step 2: Determine Multiplicity of Roots
The multiplicity of a root affects the graph's behavior at that root.
- [tex]\( x = 0 \)[/tex]: This root has multiplicity 1 (since it appears as [tex]\( x \)[/tex]).
- [tex]\( x = 3 \)[/tex]: This root has multiplicity 2 (since it appears as [tex]\( (x-3)^2 \)[/tex]).
- [tex]\( x = -2 \)[/tex]: This root has multiplicity 2 (since it appears as [tex]\( (x+2)^2 \)[/tex]).
### Step 3: End Behavior of the Polynomial
Consider the leading term to determine the polynomial's end behavior. The highest degree term in [tex]\( g(x) \)[/tex] is obtained by multiplying the leading terms in each factor:
[tex]\[ -x \cdot x^2 \cdot x^2 = -x^5 \][/tex]
As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex]:
[tex]\[ g(x) \to -x^5 \to -\infty \][/tex]
As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex]:
[tex]\[ g(x) \to -x^5 \to \infty \][/tex]
Therefore, the graph will fall to [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] goes to [tex]\( \infty \)[/tex] and rise to [tex]\( \infty \)[/tex] as [tex]\( x \)[/tex] goes to [tex]\( -\infty \)[/tex].
### Step 4: Behavior Near the Roots
- At [tex]\( x = 0 \)[/tex] (multiplicity 1): The graph crosses the x-axis.
- At [tex]\( x = 3 \)[/tex] (multiplicity 2): The graph touches the x-axis and turns around, forming a local minimum or maximum.
- At [tex]\( x = -2 \)[/tex] (multiplicity 2): The graph touches the x-axis and turns around, forming a local minimum or maximum.
### Step 5: Additional Points and Sketch
To better understand the shape of the graph, evaluate the function at additional points, particularly near the roots and around where the graph turns.
#### At [tex]\( x = -3 \)[/tex]:
[tex]\[ g(-3) = -(-3) \cdot (-3-3)^2 \cdot (-3+2)^2 = 3 \cdot 36 \cdot 1 = 108 \][/tex]
#### At [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = -(-1) \cdot (-1-3)^2 \cdot (-1+2)^2 = 1 \cdot 16 \cdot 1 = -16 \][/tex]
#### At [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = -1 \cdot (1-3)^2 \cdot (1+2)^2 = -1 \cdot 4 \cdot 9 = -36 \][/tex]
#### At [tex]\( x = 4 \)[/tex]:
[tex]\[ g(4) = -4 \cdot (4-3)^2 \cdot (4+2)^2 = -4 \cdot 1 \cdot 36 = -144 \][/tex]
### Sketch the Graph
With this information, we can sketch a rough graph:
1. The graph rises to [tex]\( \infty \)[/tex] as [tex]\( x \)[/tex] goes to [tex]\( -\infty \)[/tex].
2. It crosses the x-axis at [tex]\( x = 0 \)[/tex].
3. It touches and turns around at [tex]\( x = -2 \)[/tex] and [tex]\( x = 3 \)[/tex].
4. It has significant points at [tex]\( (-3, 108) \)[/tex], [tex]\( (-1, -16) \)[/tex], [tex]\( (1, -36) \)[/tex], and [tex]\( (4, -144) \)[/tex].
The overall shape should reflect these key features: starting from [tex]\( \infty \)[/tex] on the left, turning around at [tex]\( x = -2 \)[/tex], crossing at [tex]\( x = 0 \)[/tex], turning around again at [tex]\( x = 3 \)[/tex], and finally falling towards [tex]\( -\infty \)[/tex] on the right.
This completes the detailed examination of the function [tex]\( g(x) = -x(x-3)^2(x+2)^2 \)[/tex] and its graph.
