To determine which term, when added to the given polynomial, will change the end behavior, we need to consider the term with the highest degree. The end behavior of a polynomial function is determined by its term with the highest power of [tex]\( x \)[/tex].
The given polynomial is:
[tex]\[ y = -2x^7 + 5x^6 - 24 \][/tex]
Let's analyze each of the provided options:
1. [tex]\( -x^8 \)[/tex]
2. [tex]\( -3x^5 \)[/tex]
3. [tex]\( 5x^7 \)[/tex]
4. 1,000
5. [tex]\( -300 \)[/tex]
### Analysis:
1. Term: [tex]\( -x^8 \)[/tex]
- This term has a degree of 8, which is higher than the current highest degree term in the polynomial ([tex]\( -2x^7 \)[/tex] has a degree of 7).
- Adding [tex]\( -x^8 \)[/tex] to the polynomial will change the highest degree term, thus changing the end behavior of the polynomial.
2. Term: [tex]\( -3x^5 \)[/tex]
- This term has a degree of 5, which is lower than the current highest degree term ([tex]\( x^7 \)[/tex]).
- Adding this term will not change the polynomial's highest degree term, thus not changing the end behavior.
3. Term: [tex]\( 5x^7 \)[/tex]
- This term has the same degree (7) as the current highest degree term ([tex]\( -2x^7 \)[/tex]).
- Adding this term will not change the highest degree term's degree, thus not changing the end behavior.
4. Term: 1,000
- This term is a constant, so its degree is 0.
- Adding a constant will not affect the polynomial's highest degree term, thus not changing the end behavior.
5. Term: [tex]\( -300 \)[/tex]
- This term is also a constant (degree 0).
- Adding this term will not affect the polynomial's highest degree term, thus not changing the end behavior.
### Conclusion:
The only term that, when added to the polynomial, will change the end behavior is [tex]\( -x^8 \)[/tex]. This is because it introduces a term of a higher degree (8) than the existing highest degree term (7) in the polynomial.
Thus, the correct term to add to the polynomial to change its end behavior is:
[tex]\[ -x^8 \][/tex]
