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Answer :
Sure! Let's look at each of the options to identify which polynomial expressions are equivalent to [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex].
To determine equivalence, each expression must have the same coefficients for corresponding terms.
### Let's examine each option:
A. [tex]\(16x^{10}\)[/tex]
- This is not equivalent because the degree and terms of the polynomial are completely different from the original.
B. [tex]\(6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x\)[/tex]
- This expression has different exponents for the terms compared to the original polynomial (starts from [tex]\(x^5\)[/tex], not [tex]\(x^4\)[/tex]), which makes it not equivalent.
C. [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]
- This is exactly the same as the original polynomial, with identical terms and coefficients. Therefore, it is equivalent.
D. [tex]\(8 + 5x + 7x^2 - 4x^3 + 6x^4\)[/tex]
- Although the order is reversed, if we rearrange the terms to [tex]\(6x^4 - 4x^3 + 7x^2 + 5x + 8\)[/tex], we can see that it is not equivalent due to different signs for the [tex]\(-7x^2\)[/tex] term. Therefore, this is not equivalent.
E. [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
- If we reorder this expression to match the descending order of powers, it becomes [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex], which is identical to the original polynomial. Hence, this expression is equivalent.
### Conclusion:
The expressions that are equivalent to [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] are:
- C. [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]
- E. [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
Thus, the correct options are C and E.
To determine equivalence, each expression must have the same coefficients for corresponding terms.
### Let's examine each option:
A. [tex]\(16x^{10}\)[/tex]
- This is not equivalent because the degree and terms of the polynomial are completely different from the original.
B. [tex]\(6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x\)[/tex]
- This expression has different exponents for the terms compared to the original polynomial (starts from [tex]\(x^5\)[/tex], not [tex]\(x^4\)[/tex]), which makes it not equivalent.
C. [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]
- This is exactly the same as the original polynomial, with identical terms and coefficients. Therefore, it is equivalent.
D. [tex]\(8 + 5x + 7x^2 - 4x^3 + 6x^4\)[/tex]
- Although the order is reversed, if we rearrange the terms to [tex]\(6x^4 - 4x^3 + 7x^2 + 5x + 8\)[/tex], we can see that it is not equivalent due to different signs for the [tex]\(-7x^2\)[/tex] term. Therefore, this is not equivalent.
E. [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
- If we reorder this expression to match the descending order of powers, it becomes [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex], which is identical to the original polynomial. Hence, this expression is equivalent.
### Conclusion:
The expressions that are equivalent to [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] are:
- C. [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]
- E. [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
Thus, the correct options are C and E.
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