Which of the following terms, when added to the given polynomial, will change the end behavior?

[tex]\[ y = -2x^7 + 5x^6 - 24 \][/tex]

A. [tex]\(-x^8\)[/tex]

B. [tex]\(-3x^5\)[/tex]

C. [tex]\(5x^7\)[/tex]

D. [tex]\(1000\)[/tex]

E. [tex]\(-300\)[/tex]



Answer :

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]