Uncover valuable information and solutions with IDNLearn.com's extensive Q&A platform. Get the information you need quickly and accurately with our reliable and thorough Q&A platform.
Sagot :
To determine the end behavior of the polynomial function [tex]\( f(x) = x^5 - 8x^4 + 16x^3 \)[/tex], we need to analyze the highest degree term as [tex]\( x \)[/tex] approaches positive and negative infinity.
### Step-by-Step Solution for End Behavior
1. Identify the highest degree term:
The highest degree term in the polynomial [tex]\( f(x) = x^5 - 8x^4 + 16x^3 \)[/tex] is [tex]\( x^5 \)[/tex].
2. Analyze the end behavior as [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]):
- As [tex]\( x \to -\infty \)[/tex], the term [tex]\( x^5 \)[/tex] will dominate the behavior of the polynomial. Since [tex]\( x^5 \)[/tex] grows negatively (because raising a negative number to an odd power results in a negative number):
[tex]\[ f(x) \to -\infty \text{ as } x \to -\infty \][/tex]
3. Analyze the end behavior as [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to +\infty \)[/tex]):
- As [tex]\( x \to +\infty \)[/tex], the term [tex]\( x^5 \)[/tex] will dominate the behavior of the polynomial. Since [tex]\( x^5 \)[/tex] grows positively (because raising a positive number to an odd power results in a positive number):
[tex]\[ f(x) \to +\infty \text{ as } x \to +\infty \][/tex]
Therefore, the correct end behavior is:
[tex]\[ f(x) \to -\infty \text{ as } x \to -\infty; \quad f(x) \to +\infty \text{ as } x \to +\infty \][/tex]
### Finding where the graph touches or crosses the x-axis
To determine where the graph touches or crosses the x-axis, we need to find the roots of the polynomial and their multiplicities.
1. Factor the polynomial [tex]\( f(x) = x^5 - 8x^4 + 16x^3 \)[/tex]:
- Factor out the greatest common divisor:
[tex]\[ f(x) = x^3 (x^2 - 8x + 16) \][/tex]
- Further factorize the quadratic:
[tex]\[ x^2 - 8x + 16 = (x - 4)^2 \][/tex]
- So, the polynomial can be written as:
[tex]\[ f(x) = x^3 (x - 4)^2 \][/tex]
2. Find the roots and their multiplicities:
- The root [tex]\( x = 0 \)[/tex] has a multiplicity of 3 (since [tex]\( x^3 \)[/tex] implies three [tex]\( x \)[/tex] factors).
- The root [tex]\( x = 4 \)[/tex] has a multiplicity of 2 (since [tex]\( (x-4)^2 \)[/tex] implies two [tex]\( x-4 \)[/tex] factors).
3. Determine whether the roots touch or cross the x-axis:
- A root with odd multiplicity means the graph crosses the x-axis.
- A root with even multiplicity means the graph touches the x-axis but does not cross it.
Therefore:
[tex]\[ \text{The graph touches, but does not cross, the } x \text{-axis at } x = 4 . \][/tex]
[tex]\[ \text{The graph crosses the } x \text{-axis at } x = 0 . \][/tex]
### Final answers:
- End behavior:
[tex]\[ f(x) \to -\infty \text{ as } x \to -\infty; \quad f(x) \to +\infty \text{ as } x \to +\infty \][/tex]
- Touch the x-axis at:
[tex]\[ x = 4 \][/tex]
- Cross the x-axis at:
[tex]\[ x = 0 \][/tex]
### Step-by-Step Solution for End Behavior
1. Identify the highest degree term:
The highest degree term in the polynomial [tex]\( f(x) = x^5 - 8x^4 + 16x^3 \)[/tex] is [tex]\( x^5 \)[/tex].
2. Analyze the end behavior as [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]):
- As [tex]\( x \to -\infty \)[/tex], the term [tex]\( x^5 \)[/tex] will dominate the behavior of the polynomial. Since [tex]\( x^5 \)[/tex] grows negatively (because raising a negative number to an odd power results in a negative number):
[tex]\[ f(x) \to -\infty \text{ as } x \to -\infty \][/tex]
3. Analyze the end behavior as [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to +\infty \)[/tex]):
- As [tex]\( x \to +\infty \)[/tex], the term [tex]\( x^5 \)[/tex] will dominate the behavior of the polynomial. Since [tex]\( x^5 \)[/tex] grows positively (because raising a positive number to an odd power results in a positive number):
[tex]\[ f(x) \to +\infty \text{ as } x \to +\infty \][/tex]
Therefore, the correct end behavior is:
[tex]\[ f(x) \to -\infty \text{ as } x \to -\infty; \quad f(x) \to +\infty \text{ as } x \to +\infty \][/tex]
### Finding where the graph touches or crosses the x-axis
To determine where the graph touches or crosses the x-axis, we need to find the roots of the polynomial and their multiplicities.
1. Factor the polynomial [tex]\( f(x) = x^5 - 8x^4 + 16x^3 \)[/tex]:
- Factor out the greatest common divisor:
[tex]\[ f(x) = x^3 (x^2 - 8x + 16) \][/tex]
- Further factorize the quadratic:
[tex]\[ x^2 - 8x + 16 = (x - 4)^2 \][/tex]
- So, the polynomial can be written as:
[tex]\[ f(x) = x^3 (x - 4)^2 \][/tex]
2. Find the roots and their multiplicities:
- The root [tex]\( x = 0 \)[/tex] has a multiplicity of 3 (since [tex]\( x^3 \)[/tex] implies three [tex]\( x \)[/tex] factors).
- The root [tex]\( x = 4 \)[/tex] has a multiplicity of 2 (since [tex]\( (x-4)^2 \)[/tex] implies two [tex]\( x-4 \)[/tex] factors).
3. Determine whether the roots touch or cross the x-axis:
- A root with odd multiplicity means the graph crosses the x-axis.
- A root with even multiplicity means the graph touches the x-axis but does not cross it.
Therefore:
[tex]\[ \text{The graph touches, but does not cross, the } x \text{-axis at } x = 4 . \][/tex]
[tex]\[ \text{The graph crosses the } x \text{-axis at } x = 0 . \][/tex]
### Final answers:
- End behavior:
[tex]\[ f(x) \to -\infty \text{ as } x \to -\infty; \quad f(x) \to +\infty \text{ as } x \to +\infty \][/tex]
- Touch the x-axis at:
[tex]\[ x = 4 \][/tex]
- Cross the x-axis at:
[tex]\[ x = 0 \][/tex]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Find clear answers at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.
