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Answer :
To determine which polynomial expressions are equivalent to the expression [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex], let's analyze each option step-by-step.
### Given Expression:
The original polynomial expression we are comparing to is:
[tex]\[ 6x^4 + 4x^3 - 7x^2 + 5x + 8 \][/tex]
### Analyzing the Options:
Option A: [tex]\(16x^{10}\)[/tex]
- This is a polynomial of a much higher degree (degree 10) compared to the given polynomial (degree 4). Therefore, it is not equivalent.
Option B: [tex]\(6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x\)[/tex]
- The degrees of each term in this option are higher than or equal to those in the original expression, and the terms don't match. It's not equivalent.
Option C: [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]
- This is identical to the given expression. Therefore, it is equivalent.
Option D: [tex]\(8 + 5x + 7x^2 - 4x^3 + 6x^4\)[/tex]
- The terms are the same as the original expression, but written in reverse order. They still add up the same way, so this option is equivalent.
Option E: [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
- Similarly to Option D, the terms are reordered, but they still match the original expression’s coefficients and degrees. Therefore, this option is also equivalent.
### Conclusion:
The polynomial expressions 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 D: [tex]\(8 + 5x + 7x^2 - 4x^3 + 6x^4\)[/tex]
- Option E: [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
So, the correct selections are C, D, and E.
### Given Expression:
The original polynomial expression we are comparing to is:
[tex]\[ 6x^4 + 4x^3 - 7x^2 + 5x + 8 \][/tex]
### Analyzing the Options:
Option A: [tex]\(16x^{10}\)[/tex]
- This is a polynomial of a much higher degree (degree 10) compared to the given polynomial (degree 4). Therefore, it is not equivalent.
Option B: [tex]\(6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x\)[/tex]
- The degrees of each term in this option are higher than or equal to those in the original expression, and the terms don't match. It's not equivalent.
Option C: [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]
- This is identical to the given expression. Therefore, it is equivalent.
Option D: [tex]\(8 + 5x + 7x^2 - 4x^3 + 6x^4\)[/tex]
- The terms are the same as the original expression, but written in reverse order. They still add up the same way, so this option is equivalent.
Option E: [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
- Similarly to Option D, the terms are reordered, but they still match the original expression’s coefficients and degrees. Therefore, this option is also equivalent.
### Conclusion:
The polynomial expressions 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 D: [tex]\(8 + 5x + 7x^2 - 4x^3 + 6x^4\)[/tex]
- Option E: [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
So, the correct selections are C, D, and E.
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