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Sagot :
To determine which term will change the end behavior of the given polynomial [tex]\( y = -2x^7 + 5x^6 - 24 \)[/tex], we need to understand what dictates the end behavior of polynomials.
The end behavior of a polynomial function is determined by its highest-degree term. For the polynomial [tex]\( y = -2x^7 + 5x^6 - 24 \)[/tex], the highest-degree term is [tex]\( -2x^7 \)[/tex].
Let's analyze how each of the provided terms affects the highest-degree term:
1. Term: [tex]\( -x^8 \)[/tex]
- When [tex]\( -x^8 \)[/tex] is added to [tex]\( -2x^7 + 5x^6 - 24 \)[/tex], the new highest-degree term becomes [tex]\( -x^8 \)[/tex].
- Degree: 8 (higher than 7)
- Effect on end behavior: This term will change the end behavior of the polynomial because it introduces a new leading term with a higher degree.
2. Term: [tex]\( -3x^5 \)[/tex]
- When [tex]\( -3x^5 \)[/tex] is added to [tex]\( -2x^7 + 5x^6 - 24 \)[/tex], the highest-degree term remains [tex]\( -2x^7 \)[/tex].
- Degree: 5 (lower than 7)
- Effect on end behavior: This term will not change the end behavior as it does not affect the highest-degree term.
3. Term: [tex]\( 5x^7 \)[/tex]
- When [tex]\( 5x^7 \)[/tex] is added to [tex]\( -2x^7 + 5x^6 - 24 \)[/tex], the highest-degree term would be:
[tex]\[ -2x^7 + 5x^7 = 3x^7 \][/tex]
- Degree: 7 (same as the existing highest-degree term)
- Effect on end behavior: While it changes the coefficient of the highest-degree term, it does not change the degree itself. The end behavior, therefore, stays determined by [tex]\( x^7 \)[/tex].
4. Term: 1,000
- When [tex]\( 1,000 \)[/tex] is added to [tex]\( -2x^7 + 5x^6 - 24 \)[/tex], the highest-degree term remains [tex]\( -2x^7 \)[/tex].
- This just adds a constant term.
- Effect on end behavior: This term will not change the end behavior as it does not affect the polynomial's degree at all.
5. Term: [tex]\( -300 \)[/tex]
- When [tex]\( -300 \)[/tex] is added to [tex]\( -2x^7 + 5x^6 - 24 \)[/tex], the highest-degree term remains [tex]\( -2x^7 \)[/tex].
- This just adds another constant term.
- Effect on end behavior: This term will not change the end behavior as it does not affect the polynomial's degree at all.
Given this analysis, the term [tex]\( -x^8 \)[/tex] is the only one that changes the highest-degree term and thereby affects the end behavior of the polynomial. Therefore, the answer is:
[tex]\[ \boxed{-x^8} \][/tex]
The end behavior of a polynomial function is determined by its highest-degree term. For the polynomial [tex]\( y = -2x^7 + 5x^6 - 24 \)[/tex], the highest-degree term is [tex]\( -2x^7 \)[/tex].
Let's analyze how each of the provided terms affects the highest-degree term:
1. Term: [tex]\( -x^8 \)[/tex]
- When [tex]\( -x^8 \)[/tex] is added to [tex]\( -2x^7 + 5x^6 - 24 \)[/tex], the new highest-degree term becomes [tex]\( -x^8 \)[/tex].
- Degree: 8 (higher than 7)
- Effect on end behavior: This term will change the end behavior of the polynomial because it introduces a new leading term with a higher degree.
2. Term: [tex]\( -3x^5 \)[/tex]
- When [tex]\( -3x^5 \)[/tex] is added to [tex]\( -2x^7 + 5x^6 - 24 \)[/tex], the highest-degree term remains [tex]\( -2x^7 \)[/tex].
- Degree: 5 (lower than 7)
- Effect on end behavior: This term will not change the end behavior as it does not affect the highest-degree term.
3. Term: [tex]\( 5x^7 \)[/tex]
- When [tex]\( 5x^7 \)[/tex] is added to [tex]\( -2x^7 + 5x^6 - 24 \)[/tex], the highest-degree term would be:
[tex]\[ -2x^7 + 5x^7 = 3x^7 \][/tex]
- Degree: 7 (same as the existing highest-degree term)
- Effect on end behavior: While it changes the coefficient of the highest-degree term, it does not change the degree itself. The end behavior, therefore, stays determined by [tex]\( x^7 \)[/tex].
4. Term: 1,000
- When [tex]\( 1,000 \)[/tex] is added to [tex]\( -2x^7 + 5x^6 - 24 \)[/tex], the highest-degree term remains [tex]\( -2x^7 \)[/tex].
- This just adds a constant term.
- Effect on end behavior: This term will not change the end behavior as it does not affect the polynomial's degree at all.
5. Term: [tex]\( -300 \)[/tex]
- When [tex]\( -300 \)[/tex] is added to [tex]\( -2x^7 + 5x^6 - 24 \)[/tex], the highest-degree term remains [tex]\( -2x^7 \)[/tex].
- This just adds another constant term.
- Effect on end behavior: This term will not change the end behavior as it does not affect the polynomial's degree at all.
Given this analysis, the term [tex]\( -x^8 \)[/tex] is the only one that changes the highest-degree term and thereby affects the end behavior of the polynomial. Therefore, the answer is:
[tex]\[ \boxed{-x^8} \][/tex]
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