Answer :
Sure! Let's analyze each polynomial expression to determine if it is equivalent to [tex]\( 6x^4 + 4x^3 - 7x^2 + 5x + 8 \)[/tex].
1. The original polynomial is [tex]\( 6x^4 + 4x^3 - 7x^2 + 5x + 8 \)[/tex].
### Step-by-Step Analysis:
Option A: [tex]\( 16x^{10} \)[/tex]
- This polynomial has degree 10.
- It does not have any terms of degree 4, 3, 2, 1, or constant term that match the original polynomial.
- Conclusion: This option is not equivalent.
Option B: [tex]\( 6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x \)[/tex]
- This polynomial has different degrees compared to the original polynomial.
- The original polynomial has a highest degree of 4, while this has degree 5.
- Conclusion: This option is not equivalent.
Option C: [tex]\( 6x^4 + 4x^3 - 7x^2 + 5x + 8 \)[/tex]
- This matches exactly with the original polynomial in both terms and coefficients.
- Conclusion: This option is equivalent.
Option D: [tex]\( 8 + 5x + 7x^2 - 4x^3 + 6x^4 \)[/tex]
- We need to rewrite this polynomial in standard form: [tex]\( 6x^4 - 4x^3 + 7x^2 + 5x + 8 \)[/tex].
- Comparing the rewritten form with the original polynomial, the coefficients for [tex]\( x^3 \)[/tex] and [tex]\( x^2 \)[/tex] terms do not match.
- Conclusion: This option is not equivalent.
Option E: [tex]\( 8 + 5x - 7x^2 + 4x^3 + 6x^4 \)[/tex]
- We need to rewrite this polynomial in standard form: [tex]\( 6x^4 + 4x^3 - 7x^2 + 5x + 8 \)[/tex].
- Comparing the rewritten form with the original polynomial, all terms and coefficients match.
- Conclusion: This option is equivalent.
### Conclusion:
Based on our analysis, 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]
So, the correct answers are C and E.
1. The original polynomial is [tex]\( 6x^4 + 4x^3 - 7x^2 + 5x + 8 \)[/tex].
### Step-by-Step Analysis:
Option A: [tex]\( 16x^{10} \)[/tex]
- This polynomial has degree 10.
- It does not have any terms of degree 4, 3, 2, 1, or constant term that match the original polynomial.
- Conclusion: This option is not equivalent.
Option B: [tex]\( 6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x \)[/tex]
- This polynomial has different degrees compared to the original polynomial.
- The original polynomial has a highest degree of 4, while this has degree 5.
- Conclusion: This option is not equivalent.
Option C: [tex]\( 6x^4 + 4x^3 - 7x^2 + 5x + 8 \)[/tex]
- This matches exactly with the original polynomial in both terms and coefficients.
- Conclusion: This option is equivalent.
Option D: [tex]\( 8 + 5x + 7x^2 - 4x^3 + 6x^4 \)[/tex]
- We need to rewrite this polynomial in standard form: [tex]\( 6x^4 - 4x^3 + 7x^2 + 5x + 8 \)[/tex].
- Comparing the rewritten form with the original polynomial, the coefficients for [tex]\( x^3 \)[/tex] and [tex]\( x^2 \)[/tex] terms do not match.
- Conclusion: This option is not equivalent.
Option E: [tex]\( 8 + 5x - 7x^2 + 4x^3 + 6x^4 \)[/tex]
- We need to rewrite this polynomial in standard form: [tex]\( 6x^4 + 4x^3 - 7x^2 + 5x + 8 \)[/tex].
- Comparing the rewritten form with the original polynomial, all terms and coefficients match.
- Conclusion: This option is equivalent.
### Conclusion:
Based on our analysis, 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]
So, the correct answers are C and E.
