Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Explore our Q&A platform to find reliable answers from a wide range of experts in different fields. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To determine the nature of the intersections of the given polynomial function [tex]\( f(x) = x^5 - 6x^4 + 9x^3 \)[/tex] with the [tex]\( x \)[/tex]-axis, we need to follow these steps:
1. Find the roots of the polynomial.
2. Determine the multiplicity of each root.
3. Analyze how the graph interacts with the [tex]\( x \)[/tex]-axis based on the multiplicity of the roots.
### Step 1: Find the roots
First, set the polynomial equal to zero to find the roots:
[tex]\[ f(x) = x^5 - 6x^4 + 9x^3 = 0 \][/tex]
Factor out the common term [tex]\( x^3 \)[/tex]:
[tex]\[ x^3 (x^2 - 6x + 9) = 0 \][/tex]
Then, simplify within the parentheses:
[tex]\[ x^2 - 6x + 9 \][/tex]
Notice that this is a perfect square trinomial:
[tex]\[ x^2 - 6x + 9 = (x - 3)^2 \][/tex]
Now, the polynomial can be written as:
[tex]\[ x^3 (x - 3)^2 = 0 \][/tex]
### Step 2: Determine the multiplicity of each root
From the factorized form, we can see the roots and their multiplicities:
- [tex]\( x = 0 \)[/tex] with multiplicity 3 (from [tex]\( x^3 \)[/tex])
- [tex]\( x = 3 \)[/tex] with multiplicity 2 (from [tex]\( (x - 3)^2 \)[/tex])
### Step 3: Analyze the nature of each root
The multiplicity of a root gives us information on whether the graph crosses or touches the [tex]\( x \)[/tex]-axis at that root:
- If the multiplicity is odd, the graph crosses the [tex]\( x \)[/tex]-axis at the root.
- If the multiplicity is even, the graph touches but does not cross the [tex]\( x \)[/tex]-axis at the root.
Using these rules:
- For [tex]\( x = 0 \)[/tex] (multiplicity 3, which is odd), the graph crosses the [tex]\( x \)[/tex]-axis.
- For [tex]\( x = 3 \)[/tex] (multiplicity 2, which is even), the graph touches the [tex]\( x \)[/tex]-axis.
### Conclusion
From our analysis, the correct statement that describes the graph of the polynomial function [tex]\( f(x) = x^5 - 6x^4 + 9x^3 \)[/tex] is:
- The graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x=0 \)[/tex] and touches the [tex]\( x \)[/tex]-axis at [tex]\( x=3 \)[/tex].
So, the correct answer is:
The graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x=0 \)[/tex] and touches the [tex]\( x \)[/tex]-axis at [tex]\( x=3 \)[/tex].
1. Find the roots of the polynomial.
2. Determine the multiplicity of each root.
3. Analyze how the graph interacts with the [tex]\( x \)[/tex]-axis based on the multiplicity of the roots.
### Step 1: Find the roots
First, set the polynomial equal to zero to find the roots:
[tex]\[ f(x) = x^5 - 6x^4 + 9x^3 = 0 \][/tex]
Factor out the common term [tex]\( x^3 \)[/tex]:
[tex]\[ x^3 (x^2 - 6x + 9) = 0 \][/tex]
Then, simplify within the parentheses:
[tex]\[ x^2 - 6x + 9 \][/tex]
Notice that this is a perfect square trinomial:
[tex]\[ x^2 - 6x + 9 = (x - 3)^2 \][/tex]
Now, the polynomial can be written as:
[tex]\[ x^3 (x - 3)^2 = 0 \][/tex]
### Step 2: Determine the multiplicity of each root
From the factorized form, we can see the roots and their multiplicities:
- [tex]\( x = 0 \)[/tex] with multiplicity 3 (from [tex]\( x^3 \)[/tex])
- [tex]\( x = 3 \)[/tex] with multiplicity 2 (from [tex]\( (x - 3)^2 \)[/tex])
### Step 3: Analyze the nature of each root
The multiplicity of a root gives us information on whether the graph crosses or touches the [tex]\( x \)[/tex]-axis at that root:
- If the multiplicity is odd, the graph crosses the [tex]\( x \)[/tex]-axis at the root.
- If the multiplicity is even, the graph touches but does not cross the [tex]\( x \)[/tex]-axis at the root.
Using these rules:
- For [tex]\( x = 0 \)[/tex] (multiplicity 3, which is odd), the graph crosses the [tex]\( x \)[/tex]-axis.
- For [tex]\( x = 3 \)[/tex] (multiplicity 2, which is even), the graph touches the [tex]\( x \)[/tex]-axis.
### Conclusion
From our analysis, the correct statement that describes the graph of the polynomial function [tex]\( f(x) = x^5 - 6x^4 + 9x^3 \)[/tex] is:
- The graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x=0 \)[/tex] and touches the [tex]\( x \)[/tex]-axis at [tex]\( x=3 \)[/tex].
So, the correct answer is:
The graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x=0 \)[/tex] and touches the [tex]\( x \)[/tex]-axis at [tex]\( x=3 \)[/tex].
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.