Answer :
Sure, let's go through each option step-by-step to determine which polynomial expressions are equivalent to [tex]\(6 x^4 + 4 x^3 - 7 x^2 + 5 x + 8\)[/tex].
1. First polynomial: [tex]\(16 x^{10}\)[/tex]
- This polynomial is not equivalent because the degrees and coefficients of the terms are different.
2. Second polynomial: [tex]\(6 x^5 + 4 x^4 - 7 x^3 + 5 x^2 + 8 x\)[/tex]
- This polynomial has different degrees in several terms (e.g., [tex]\(x^5\)[/tex] instead of [tex]\(x^4\)[/tex]), so it is not equivalent.
3. Third polynomial: [tex]\(6 x^4 + 4 x^3 - 7 x^2 + 5 x + 8\)[/tex]
- This polynomial matches exactly with the given polynomial. Therefore, it is equivalent.
4. Fourth polynomial: [tex]\(8 + 5 x + 7 x^2 - 4 x^3 + 6 x^4\)[/tex]
- Rearranging the terms in descending order of their degrees gives [tex]\(6 x^4 - 4 x^3 + 7 x^2 + 5 x + 8\)[/tex]. The coefficients for [tex]\(x^3\)[/tex] and [tex]\(x^2\)[/tex] are different, so it is not equivalent.
5. Fifth polynomial: [tex]\(8 + 5 x - 7 x^2 + 4 x^3 + 6 x^4\)[/tex]
- Rearranging the terms in descending order of their degrees gives [tex]\(6 x^4 + 4 x^3 - 7 x^2 + 5 x + 8\)[/tex]. The arrangement matches the given polynomial, so it is equivalent.
Therefore, the polynomial expressions that are equivalent to [tex]\(6 x^4 + 4 x^3 - 7 x^2 + 5 x + 8\)[/tex] are:
- [tex]\(6 x^4 + 4 x^3 - 7 x^2 + 5 x + 8\)[/tex]
- [tex]\(8 + 5 x - 7 x^2 + 4 x^3 + 6 x^4\)[/tex]
These are the third and fifth expressions in the list.
1. First polynomial: [tex]\(16 x^{10}\)[/tex]
- This polynomial is not equivalent because the degrees and coefficients of the terms are different.
2. Second polynomial: [tex]\(6 x^5 + 4 x^4 - 7 x^3 + 5 x^2 + 8 x\)[/tex]
- This polynomial has different degrees in several terms (e.g., [tex]\(x^5\)[/tex] instead of [tex]\(x^4\)[/tex]), so it is not equivalent.
3. Third polynomial: [tex]\(6 x^4 + 4 x^3 - 7 x^2 + 5 x + 8\)[/tex]
- This polynomial matches exactly with the given polynomial. Therefore, it is equivalent.
4. Fourth polynomial: [tex]\(8 + 5 x + 7 x^2 - 4 x^3 + 6 x^4\)[/tex]
- Rearranging the terms in descending order of their degrees gives [tex]\(6 x^4 - 4 x^3 + 7 x^2 + 5 x + 8\)[/tex]. The coefficients for [tex]\(x^3\)[/tex] and [tex]\(x^2\)[/tex] are different, so it is not equivalent.
5. Fifth polynomial: [tex]\(8 + 5 x - 7 x^2 + 4 x^3 + 6 x^4\)[/tex]
- Rearranging the terms in descending order of their degrees gives [tex]\(6 x^4 + 4 x^3 - 7 x^2 + 5 x + 8\)[/tex]. The arrangement matches the given polynomial, so it is equivalent.
Therefore, the polynomial expressions that are equivalent to [tex]\(6 x^4 + 4 x^3 - 7 x^2 + 5 x + 8\)[/tex] are:
- [tex]\(6 x^4 + 4 x^3 - 7 x^2 + 5 x + 8\)[/tex]
- [tex]\(8 + 5 x - 7 x^2 + 4 x^3 + 6 x^4\)[/tex]
These are the third and fifth expressions in the list.
