Answer :
Sure! Let's examine each polynomial to determine which ones are equivalent to the given polynomial [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex].
### Given Polynomial:
[tex]\[6x^4 + 4x^3 - 7x^2 + 5x + 8\][/tex]
### Option A:
[tex]\[16x^{10}\][/tex]
This polynomial has a single term with [tex]\(x^{10}\)[/tex], and it does not match the given polynomial, which has terms of [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and a constant. Therefore, this is not equivalent.
### Option B:
[tex]\[6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x\][/tex]
This polynomial has terms with degrees up to 5, while the given polynomial has the highest degree of 4. Thus, this polynomial is not equivalent.
### Option C:
[tex]\[6x^4 + 4x^3 - 7x^2 + 5x + 8\][/tex]
This polynomial exactly matches the given polynomial in every term. Therefore, this polynomial is equivalent.
### Option D:
[tex]\[8 + 5x + 7x^2 - 4x^3 + 6x^4\][/tex]
This polynomial can be rearranged:
[tex]\[6x^4 - 4x^3 + 7x^2 + 5x + 8\][/tex]
When we look closely, the signs of the coefficients of the [tex]\(x^3\)[/tex] and [tex]\(x^2\)[/tex] terms are opposite compared to the given polynomial. Hence, it is not equivalent.
### Option E:
[tex]\[8 + 5x - 7x^2 + 4x^3 + 6x^4\][/tex]
This polynomial can be rearranged as:
[tex]\[6x^4 + 4x^3 - 7x^2 + 5x + 8\][/tex]
This rearrangement matches the given polynomial exactly. Therefore, this polynomial is equivalent.
### Conclusion:
The polynomials equivalent to [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] are:
- [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] (Option C)
- [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex] (Option E)
Thus, the selected options are:
- Option C
- Option E
### Given Polynomial:
[tex]\[6x^4 + 4x^3 - 7x^2 + 5x + 8\][/tex]
### Option A:
[tex]\[16x^{10}\][/tex]
This polynomial has a single term with [tex]\(x^{10}\)[/tex], and it does not match the given polynomial, which has terms of [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and a constant. Therefore, this is not equivalent.
### Option B:
[tex]\[6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x\][/tex]
This polynomial has terms with degrees up to 5, while the given polynomial has the highest degree of 4. Thus, this polynomial is not equivalent.
### Option C:
[tex]\[6x^4 + 4x^3 - 7x^2 + 5x + 8\][/tex]
This polynomial exactly matches the given polynomial in every term. Therefore, this polynomial is equivalent.
### Option D:
[tex]\[8 + 5x + 7x^2 - 4x^3 + 6x^4\][/tex]
This polynomial can be rearranged:
[tex]\[6x^4 - 4x^3 + 7x^2 + 5x + 8\][/tex]
When we look closely, the signs of the coefficients of the [tex]\(x^3\)[/tex] and [tex]\(x^2\)[/tex] terms are opposite compared to the given polynomial. Hence, it is not equivalent.
### Option E:
[tex]\[8 + 5x - 7x^2 + 4x^3 + 6x^4\][/tex]
This polynomial can be rearranged as:
[tex]\[6x^4 + 4x^3 - 7x^2 + 5x + 8\][/tex]
This rearrangement matches the given polynomial exactly. Therefore, this polynomial is equivalent.
### Conclusion:
The polynomials equivalent to [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] are:
- [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] (Option C)
- [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex] (Option E)
Thus, the selected options are:
- Option C
- Option E
