Answer :
Sure! Let's carefully evaluate each option to determine which polynomial expressions are equivalent to [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex].
We'll check each option one by one:
- Option A: [tex]\(16x^{10}\)[/tex]
This expression doesn't match the degree or the terms of the polynomial [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]. The degree here is 10, which is not the same as 4. So, this is not equivalent.
- Option B: [tex]\(6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x\)[/tex]
Again, the terms here do not match the original polynomial. The degree in this expression is 5 instead of 4, and the coefficients of the terms are not in the correct sequence as in the given polynomial. This option is not equivalent.
- Option C: [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]
This is exactly the same as the original polynomial [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]. It is equivalent.
- Option D: [tex]\(8 + 5x + 7x^2 - 4x^3 + 6x^4\)[/tex]
Let's rewrite this in standard polynomial form, ordering terms from the highest degree to the lowest: [tex]\(6x^4 - 4x^3 + 7x^2 + 5x + 8\)[/tex]. Upon comparison, notice the change in sign for the [tex]\(x^3\)[/tex] and [tex]\(x^2\)[/tex] terms; thus, this option is not equivalent.
- Option E: [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
Rearranging this in standard polynomial form, it becomes [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]. This matches exactly with the polynomial [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] given in the question. This is also equivalent.
So, the polynomial expressions that are equivalent to [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] are:
- Option C: [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]
- Option E: [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
The correct answers are C and E.
We'll check each option one by one:
- Option A: [tex]\(16x^{10}\)[/tex]
This expression doesn't match the degree or the terms of the polynomial [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]. The degree here is 10, which is not the same as 4. So, this is not equivalent.
- Option B: [tex]\(6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x\)[/tex]
Again, the terms here do not match the original polynomial. The degree in this expression is 5 instead of 4, and the coefficients of the terms are not in the correct sequence as in the given polynomial. This option is not equivalent.
- Option C: [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]
This is exactly the same as the original polynomial [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]. It is equivalent.
- Option D: [tex]\(8 + 5x + 7x^2 - 4x^3 + 6x^4\)[/tex]
Let's rewrite this in standard polynomial form, ordering terms from the highest degree to the lowest: [tex]\(6x^4 - 4x^3 + 7x^2 + 5x + 8\)[/tex]. Upon comparison, notice the change in sign for the [tex]\(x^3\)[/tex] and [tex]\(x^2\)[/tex] terms; thus, this option is not equivalent.
- Option E: [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
Rearranging this in standard polynomial form, it becomes [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]. This matches exactly with the polynomial [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] given in the question. This is also equivalent.
So, the polynomial expressions that are equivalent to [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] are:
- Option C: [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]
- Option E: [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
The correct answers are C and E.
