Let's analyze the end behavior of the polynomial function [tex]\( p(x) = 6x^9 + 4x^6 + 2x^4 - 200 \)[/tex].
To determine the end behavior of the polynomial, we should focus on the term with the highest degree because it will dominate the behavior of the polynomial as [tex]\( x \)[/tex] approaches positive or negative infinity.
The highest degree term in this polynomial is [tex]\( 6x^9 \)[/tex].
- As [tex]\( x \rightarrow \infty \)[/tex]:
The term [tex]\( 6x^9 \)[/tex] will dominate, and since the coefficient (6) is positive and the exponent (9) is odd, [tex]\( x^9 \)[/tex] also grows without bound as [tex]\( x \)[/tex] increases. Therefore, [tex]\( 6x^9 \rightarrow \infty \)[/tex] as [tex]\( x \rightarrow \infty \)[/tex]. Hence, [tex]\( p(x) \rightarrow \infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex]:
Again, the term [tex]\( 6x^9 \)[/tex] will dominate. Since the exponent (9) is odd, [tex]\( (-x)^9 = -x^9 \)[/tex]; thus [tex]\( 6x^9 \rightarrow -\infty \)[/tex] as [tex]\( x \rightarrow -\infty \)[/tex]. Hence, [tex]\( p(x) \rightarrow -\infty \)[/tex].
Thus, the end behavior of the polynomial [tex]\( p(x) = 6x^9 + 4x^6 + 2x^4 - 200 \)[/tex] is:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( p(x) \rightarrow \infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( p(x) \rightarrow -\infty \)[/tex].
Therefore, the correct answer is:
(D) As [tex]\( x \rightarrow \infty, p(x) \rightarrow \infty \)[/tex], and as [tex]\( x \rightarrow -\infty, p(x) \rightarrow -\infty \)[/tex].
