To determine which term, when added to the polynomial [tex]\( y = -2x^7 + 5x^6 - 24 \)[/tex], will change the end behavior, we need to understand the concept of end behavior in polynomials.
End behavior of a polynomial is determined by the term with the highest degree. This is because as [tex]\( |x| \)[/tex] becomes very large, the term with the highest degree grows much faster than the other terms and thus dominates the behavior of the polynomial.
Given polynomial:
[tex]\[ y = -2x^7 + 5x^6 - 24 \][/tex]
The term with the highest degree here is [tex]\( -2x^7 \)[/tex], which means the end behavior is dictated by this term.
Let’s analyze each of the given options to see if they will change the end behavior:
1. [tex]\( -x^8 \)[/tex]:
- This term has a higher degree than [tex]\( -2x^7 \)[/tex]. Since [tex]\( x^8 \)[/tex] grows faster than [tex]\( x^7 \)[/tex] for large [tex]\( |x| \)[/tex], adding this term will dominate the behavior of the polynomial as [tex]\( x \)[/tex] grows larger. Thus, this term will change the end behavior.
2. [tex]\( -3x^5 \)[/tex]:
- This term has a lower degree than [tex]\( -2x^7 \)[/tex]. Terms with lower degrees do not affect the end behavior of the polynomial. Therefore, this will not change the end behavior.
3. [tex]\( 5x^7 \)[/tex]:
- This term has the same degree as [tex]\( -2x^7 \)[/tex]. Adding this term will change the coefficient of the [tex]\( x^7 \)[/tex] term, but it will still be dominated by [tex]\( x^7 \)[/tex]. Thus, the end behavior remains fundamentally the same and will not change.
4. [tex]\( 1,000 \)[/tex]:
- This is a constant term and has no [tex]\( x \)[/tex] dependency. Constant terms do not affect the end behavior since they do not grow as [tex]\( x \)[/tex] becomes large. Hence, this term will not change the end behavior.
5. [tex]\( -300 \)[/tex]:
- Similar to the constant term [tex]\( 1,000 \)[/tex], this also does not affect the end behavior for the same reasons. Therefore, it will not change the end behavior.
Conclusion: The term [tex]\( -x^8 \)[/tex] when added to the polynomial [tex]\( y = -2x^7 + 5x^6 - 24 \)[/tex], will change the end behavior.
Thus, the term that will change the end behavior of the given polynomial is:
[tex]\[ \boxed{-x^8} \][/tex]
