To determine how the graph of the polynomial [tex]\( y = 8x^4 - 2x^3 + 5 \)[/tex] changes when the term [tex]\( 2x^5 \)[/tex] is added, we need to examine the behavior of the new polynomial [tex]\( y = 2x^5 + 8x^4 - 2x^3 + 5 \)[/tex].
### Step-by-Step Solution:
1. Identify the dominant term:
- The highest degree term in the polynomial [tex]\( y = 2x^5 + 8x^4 - 2x^3 + 5 \)[/tex] is [tex]\( 2x^5 \)[/tex].
- The dominant term [tex]\( 2x^5 \)[/tex] will determine the end behavior of the polynomial for very large positive or negative values of [tex]\( x \)[/tex].
2. Determine the end behavior for large positive [tex]\( x \)[/tex]:
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( x^5 \)[/tex] also approaches positive infinity.
- Since [tex]\( 2x^5 \)[/tex] has a positive coefficient [tex]\( (2) \)[/tex], the value of [tex]\( y \)[/tex] will approach positive infinity.
3. Determine the end behavior for large negative [tex]\( x \)[/tex]:
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( x^5 \)[/tex] will approach negative infinity (since raising a negative number to an odd power results in a negative number).
- Since [tex]\( 2x^5 \)[/tex] has a positive coefficient [tex]\( (2) \)[/tex], multiplying it by a very large negative number will still result in [tex]\( y \)[/tex] approaching negative infinity.
4. Conclusion:
- From our analysis, when [tex]\( x \)[/tex] is very large and positive, [tex]\( y \)[/tex] approaches positive infinity.
- When [tex]\( x \)[/tex] is very large and negative, [tex]\( y \)[/tex] approaches negative infinity.
- Therefore, the ends of the graph will extend in opposite directions.
Thus, the correct statement is:
The ends of the graph will extend in opposite directions.
