To determine the end behavior of the polynomial function [tex]\( q(x) = -2 x^8 + 5 x^6 - 3 x^5 + 50 \)[/tex], we need to analyze the leading term of the polynomial. The leading term is the term with the highest degree. This term will dictate the end behavior of the polynomial as [tex]\( x \to \infty \)[/tex] and [tex]\( x \to -\infty \)[/tex].
### Step-by-Step Analysis:
1. Identify the leading term:
The polynomial is given as [tex]\( q(x) = -2 x^8 + 5 x^6 - 3 x^5 + 50 \)[/tex]. The leading term here is [tex]\( -2 x^8 \)[/tex], as it has the highest power of [tex]\( x \)[/tex] which is 8.
2. Determine the degree of the polynomial:
The degree of the polynomial is determined by the highest power of [tex]\( x \)[/tex], which is 8.
3. Determine the leading coefficient:
The coefficient of the leading term [tex]\( -2 x^8 \)[/tex] is [tex]\(-2\)[/tex].
4. Analyze the leading term for the end behavior:
- Since the degree (8) is even, the end behavior will be influenced similarly for both [tex]\( x \to \infty \)[/tex] and [tex]\( x \to -\infty \)[/tex].
- Since the leading coefficient [tex]\(-2\)[/tex] is negative, the polynomial will go to negative infinity as [tex]\( x \)[/tex] moves towards positive or negative infinity.
5. Conclude the end behavior:
- As [tex]\( x \to \infty \)[/tex], [tex]\( -2 x^8 \to -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( -2 x^8 \to -\infty \)[/tex].
Therefore, the correct end behavior of the graph of [tex]\( q(x) = -2 x^8 + 5 x^6 - 3 x^5 + 50 \)[/tex] is represented by option (C):
- As [tex]\( x \to \infty, q(x) \to -\infty \)[/tex].
- As [tex]\( x \to -\infty, q(x) \to -\infty \)[/tex].
Thus, the correct answer is:
(C) As [tex]\( x \rightarrow \infty, q(x) \rightarrow -\infty \)[/tex], and as [tex]\( x \rightarrow -\infty, q(x) \rightarrow -\infty \)[/tex].
