Answer :
To determine which terms added to the polynomial [tex]\( f(x) = 8x^3 - 5x^6 + 2x^2 - 24 \)[/tex] will change the end behavior, let's understand the concept of end behavior:
1. End Behavior of Polynomials: The end behavior of a polynomial is determined by its highest degree term. For the polynomial [tex]\( f(x) = 8x^3 - 5x^6 + 2x^2 - 24 \)[/tex], the highest degree term is [tex]\( -5x^6 \)[/tex].
2. Effect of Different Terms on End Behavior:
- If you add a term with a degree lower than 6, it will not change the end behavior because it will not surpass the effect of the original highest degree term.
- If you add a term with the same or higher degree as the existing highest degree term, it can potentially change the end behavior because it can either enhance or override the existing highest power term.
3. Analyzing Each Option:
- [tex]\(-10x^3\)[/tex]: This term has a degree of 3, which is lower than 6, so it does not change the end behavior.
- [tex]\(3x^6\)[/tex]: This term has the same degree of 6, which can influence or change the end behavior of the polynomial.
- [tex]\(8x^6\)[/tex]: This term also has the same degree of 6, so it can alter the end behavior.
- 100: This is a constant term with a degree of 0, which will not affect the end behavior.
Therefore, the terms that can change the end behavior of the given polynomial are [tex]\( 3x^6 \)[/tex] and [tex]\( 8x^6 \)[/tex].
1. End Behavior of Polynomials: The end behavior of a polynomial is determined by its highest degree term. For the polynomial [tex]\( f(x) = 8x^3 - 5x^6 + 2x^2 - 24 \)[/tex], the highest degree term is [tex]\( -5x^6 \)[/tex].
2. Effect of Different Terms on End Behavior:
- If you add a term with a degree lower than 6, it will not change the end behavior because it will not surpass the effect of the original highest degree term.
- If you add a term with the same or higher degree as the existing highest degree term, it can potentially change the end behavior because it can either enhance or override the existing highest power term.
3. Analyzing Each Option:
- [tex]\(-10x^3\)[/tex]: This term has a degree of 3, which is lower than 6, so it does not change the end behavior.
- [tex]\(3x^6\)[/tex]: This term has the same degree of 6, which can influence or change the end behavior of the polynomial.
- [tex]\(8x^6\)[/tex]: This term also has the same degree of 6, so it can alter the end behavior.
- 100: This is a constant term with a degree of 0, which will not affect the end behavior.
Therefore, the terms that can change the end behavior of the given polynomial are [tex]\( 3x^6 \)[/tex] and [tex]\( 8x^6 \)[/tex].
