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Answer :
To determine which polynomial expressions are equivalent to the given polynomial [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex], we need to compare each provided option with the given polynomial.
Let's analyze each option:
1. Option A: [tex]\(16x^{10}\)[/tex]
This expression has an entirely different degree and format compared to the given polynomial. It is not equivalent.
2. Option B: [tex]\(6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x\)[/tex]
This expression has different exponents for each term compared to the given polynomial. It is not equivalent.
3. Option C: [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]
This expression is exactly the same as the given polynomial. Therefore, it is equivalent.
4. Option D: [tex]\(8 + 5x + 7x^2 - 4x^3 + 6x^4\)[/tex]
Although this expression contains all the same terms, they are in a different order. Remember that polynomials are often reordered in standard form with terms from highest to lowest degree. Reordering this expression gives:
[tex]\( 6x^4 - 4x^3 + 7x^2 + 5x + 8 \)[/tex]
This does not match the given polynomial because signs of some terms are different. So, it is not equivalent.
5. Option E: [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
Similarly, this expression also contains all the same terms but reordered. Let's reorder:
[tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]
This matches exactly with the given polynomial. Therefore, this expression is equivalent.
Therefore, based on the comparisons, 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]
Thus, the correct answers are:
- Option C
- Option E
Let's analyze each option:
1. Option A: [tex]\(16x^{10}\)[/tex]
This expression has an entirely different degree and format compared to the given polynomial. It is not equivalent.
2. Option B: [tex]\(6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x\)[/tex]
This expression has different exponents for each term compared to the given polynomial. It is not equivalent.
3. Option C: [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]
This expression is exactly the same as the given polynomial. Therefore, it is equivalent.
4. Option D: [tex]\(8 + 5x + 7x^2 - 4x^3 + 6x^4\)[/tex]
Although this expression contains all the same terms, they are in a different order. Remember that polynomials are often reordered in standard form with terms from highest to lowest degree. Reordering this expression gives:
[tex]\( 6x^4 - 4x^3 + 7x^2 + 5x + 8 \)[/tex]
This does not match the given polynomial because signs of some terms are different. So, it is not equivalent.
5. Option E: [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
Similarly, this expression also contains all the same terms but reordered. Let's reorder:
[tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]
This matches exactly with the given polynomial. Therefore, this expression is equivalent.
Therefore, based on the comparisons, 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]
Thus, the correct answers are:
- Option C
- Option E
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