To determine the end behavior of the polynomial function [tex]\( g(x) = 5x^6 + x^5 + 9x^3 - 12x - 125 \)[/tex], we need to focus on the leading term of the polynomial, which is the term with the highest degree.
1. Identify the Leading Term: The leading term of the polynomial [tex]\( g(x) = 5x^6 + x^5 + 9x^3 - 12x - 125 \)[/tex] is the term with the highest power of [tex]\( x \)[/tex]. In this case, it is [tex]\( 5x^6 \)[/tex] since the highest exponent is 6.
2. Analyze the Leading Term: The leading term, [tex]\( 5x^6 \)[/tex], will dominate the behavior of the polynomial as [tex]\( x \)[/tex] approaches very large positive or negative values because the highest power term grows faster than any other terms.
3. Behavior as [tex]\( x \)[/tex] Approaches Positive Infinity:
- When [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex]:
[tex]\[
5x^6 \rightarrow +\infty
\][/tex]
Since the leading term [tex]\( 5x^6 \)[/tex] has a positive coefficient (5) and the exponent (6) is even, [tex]\( 5x^6 \)[/tex] will approach [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] becomes very large in the positive direction.
4. Behavior as [tex]\( x \)[/tex] Approaches Negative Infinity:
- When [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex]:
[tex]\[
5x^6 \rightarrow +\infty
\][/tex]
Even though [tex]\( x \)[/tex] is negative, because the exponent (6) is even, raising a negative number to an even power results in a positive value. Therefore, [tex]\( 5x^6 \)[/tex] will also approach [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] becomes very large in the negative direction.
By focusing on the leading term [tex]\( 5x^6 \)[/tex] and considering its coefficient and exponent, we can determine the end behavior of the graph of [tex]\( g(x) \)[/tex]:
- As [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], [tex]\( g(x) \)[/tex] approaches [tex]\( +\infty \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( g(x) \)[/tex] also approaches [tex]\( +\infty \)[/tex].
Hence, the end behavior of the graph of [tex]\( g(x) = 5x^6 + x^5 + 9x^3 - 12x - 125 \)[/tex] is:
[tex]\[
+\infty \text{ when } x \rightarrow +\infty, \quad +\infty \text{ when } x \rightarrow -\infty
\][/tex]
