To determine the end behavior of the polynomial function [tex]\( q(x) = -2x^8 + 5x^6 - 3x^5 + 50 \)[/tex], we need to look at the highest degree term, which is [tex]\( -2x^8 \)[/tex]. The leading term is the most significant term for determining the end behavior as [tex]\( x \)[/tex] becomes very large (positively or negatively).
For the polynomial [tex]\( q(x) = -2x^8 + 5x^6 - 3x^5 + 50 \)[/tex]:
1. Identify the leading term: The leading term is [tex]\( -2x^8 \)[/tex].
2. Determine the coefficient and power of the leading term:
- The coefficient is [tex]\(-2\)[/tex] (negative) and the power is [tex]\(8\)[/tex] (even).
3. Analyze the end behavior based on the leading term:
- Because the power is even ([tex]\(8\)[/tex]), [tex]\(x^8\)[/tex] will always yield positive values for large positive or negative [tex]\(x\)[/tex].
- Multiplying by the negative coefficient [tex]\(-2\)[/tex] will result in negative values.
Therefore, as [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \rightarrow \infty \)[/tex]), the term [tex]\( -2x^8 \)[/tex] will lead [tex]\( q(x) \)[/tex] towards negative infinity ([tex]\( q(x) \rightarrow -\infty \)[/tex]).
Similarly, as [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \rightarrow -\infty \)[/tex]), the term [tex]\( -2x^8 \)[/tex] will also lead [tex]\( q(x) \)[/tex] towards negative infinity ([tex]\( q(x) \rightarrow -\infty \)[/tex]).
Thus, the correct answer is:
(c) As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( q(x) \rightarrow -\infty \)[/tex], and as [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( q(x) \rightarrow -\infty \)[/tex].
