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Sagot :
To determine the degree and leading coefficient of a polynomial function based on its end behavior, we need to analyze the behavior of the polynomial as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex] and [tex]\( +\infty \)[/tex].
1. Understanding End Behavior:
The end behavior of a polynomial function is primarily influenced by its highest degree term, specifically the degree of the term and its leading coefficient. Here's a breakdown of how the degree and leading coefficient affect the end behavior:
- Odd Degree:
- Positive Leading Coefficient: The function [tex]\( f(x) \)[/tex] approaches [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], and [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex].
- Negative Leading Coefficient: The function [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], and [tex]\( f(x) \)[/tex] approaches [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex].
- Even Degree:
- Positive Leading Coefficient: The function [tex]\( f(x) \)[/tex] approaches [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] approaches both [tex]\( +\infty \)[/tex] and [tex]\( -\infty \)[/tex].
- Negative Leading Coefficient: The function [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches both [tex]\( +\infty \)[/tex] and [tex]\( -\infty \)[/tex].
2. Analysis of Given Behavior:
For the function [tex]\( f(x) \)[/tex] described in the question, it approaches [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex] and [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex]. This indicates that:
- The polynomial function must be of an odd degree because the end behaviors are in opposite directions.
- The leading coefficient must be negative because as [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \to +\infty \)[/tex], and as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
3. Matching with Given Options:
- Degree 3, leading coefficient -1: This matches the conditions because 3 is an odd degree, and the leading coefficient is negative.
- Degree 4, leading coefficient 1: This does not match because 4 is an even degree.
- Degree 5, leading coefficient 1: This does not match because the leading coefficient is positive.
- Degree 6, leading coefficient -1: This does not match because 6 is an even degree.
Therefore, the degree and leading coefficient of the polynomial function must be:
[tex]\[ \boxed{3, -1} \][/tex]
1. Understanding End Behavior:
The end behavior of a polynomial function is primarily influenced by its highest degree term, specifically the degree of the term and its leading coefficient. Here's a breakdown of how the degree and leading coefficient affect the end behavior:
- Odd Degree:
- Positive Leading Coefficient: The function [tex]\( f(x) \)[/tex] approaches [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], and [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex].
- Negative Leading Coefficient: The function [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], and [tex]\( f(x) \)[/tex] approaches [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex].
- Even Degree:
- Positive Leading Coefficient: The function [tex]\( f(x) \)[/tex] approaches [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] approaches both [tex]\( +\infty \)[/tex] and [tex]\( -\infty \)[/tex].
- Negative Leading Coefficient: The function [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches both [tex]\( +\infty \)[/tex] and [tex]\( -\infty \)[/tex].
2. Analysis of Given Behavior:
For the function [tex]\( f(x) \)[/tex] described in the question, it approaches [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex] and [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex]. This indicates that:
- The polynomial function must be of an odd degree because the end behaviors are in opposite directions.
- The leading coefficient must be negative because as [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \to +\infty \)[/tex], and as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
3. Matching with Given Options:
- Degree 3, leading coefficient -1: This matches the conditions because 3 is an odd degree, and the leading coefficient is negative.
- Degree 4, leading coefficient 1: This does not match because 4 is an even degree.
- Degree 5, leading coefficient 1: This does not match because the leading coefficient is positive.
- Degree 6, leading coefficient -1: This does not match because 6 is an even degree.
Therefore, the degree and leading coefficient of the polynomial function must be:
[tex]\[ \boxed{3, -1} \][/tex]
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