To determine the end behavior of the graph of the polynomial [tex]\( g(x) = 5x^6 + x^5 + 9x^3 - 12x - 125 \)[/tex], we focus on the term with the highest power of [tex]\( x \)[/tex], because as [tex]\( x \)[/tex] becomes very large (positively or negatively), this term will dominate the behavior of the polynomial.
1. Identify the leading term:
[tex]\[
5x^6
\][/tex]
2. Consider the leading term for large values of [tex]\( x \)[/tex]:
- When [tex]\( x \to \infty \)[/tex], the term [tex]\( 5x^6 \)[/tex] will dominate and since the coefficient 5 is positive, [tex]\( 5x^6 \to \infty \)[/tex].
- When [tex]\( x \to -\infty \)[/tex], the term [tex]\( 5x^6 \)[/tex] will also dominate. Because the power [tex]\( 6 \)[/tex] is even, [tex]\( (-x)^6 = x^6 \)[/tex], and thus [tex]\( 5(-x)^6 = 5x^6 \)[/tex]. Since the coefficient is positive, [tex]\( 5x^6 \to \infty \)[/tex].
3. Conclusion:
- As [tex]\( x \to \infty \)[/tex], [tex]\( g(x) \to \infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( g(x) \to \infty \)[/tex].
Therefore, the correct choice that describes the end behavior of the graph of [tex]\( g(x) \)[/tex] is:
(A) As [tex]\( x \to \infty, g(x) \to \infty \)[/tex], and as [tex]\( x \to -\infty, g(x) \to \infty \)[/tex].
