The graph of a polynomial function approaches [tex]$-\infty$[/tex] as [tex]$x$[/tex] approaches [tex]$-\infty$[/tex], and approaches [tex]$+\infty$[/tex] as [tex]$x$[/tex] approaches [tex]$+\infty$[/tex]. Which could be the degree and leading coefficient of this function?

A. degree 3, leading coefficient -1
B. degree 4, leading coefficient 1
C. degree 5, leading coefficient 1
D. degree 6, leading coefficient -1



Answer :

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]