Answer :
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]
