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Answer :
Sure, let's break down the process to determine which polynomial expressions are equivalent to [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex].
### Step 1: Understand the Original Polynomial
The polynomial we need to match is:
[tex]\[ 6x^4 + 4x^3 - 7x^2 + 5x + 8 \][/tex]
### Step 2: Analyze Each Choice
Choice A: [tex]\(16x^{10}\)[/tex]
- This is a single term polynomial and is clearly not in the form of the original polynomial, which has multiple terms.
- Therefore, [tex]\(16x^{10}\)[/tex] is not equivalent to the original polynomial.
Choice B: [tex]\(6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x\)[/tex]
- Compare the terms of this polynomial to the original:
- The degree of the leading term here is [tex]\(x^5\)[/tex], whereas our original polynomial has a leading term of [tex]\(x^4\)[/tex].
- Additionally, the powers and coefficients of the subsequent terms do not match the original polynomial.
- Therefore, [tex]\(6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x\)[/tex] is not equivalent to the original polynomial.
Choice C: [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]
- This exactly matches our original polynomial term for term.
- Therefore, [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] is equivalent to the original polynomial.
Choice D: [tex]\(8 + 5x + 7x^2 - 4x^3 + 6x^4\)[/tex]
- Let's reorder this polynomial to compare more easily with the original:
- Ordered polynomial is [tex]\(6x^4 - 4x^3 + 7x^2 + 5x + 8\)[/tex].
- Although the independent term and linear term are the same, the signs and coefficients of other terms do not match.
- Therefore, [tex]\(8 + 5x + 7x^2 - 4x^3 + 6x^4\)[/tex] is not equivalent to the original polynomial.
Choice E: [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
- Let's reorder this polynomial to compare more easily with the original:
- Ordered polynomial is [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex].
- This exactly matches our original polynomial term for term.
- Therefore, [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex] is equivalent to the original polynomial.
### Step 3: Conclusion
Based on our detailed comparisons, the polynomial expressions that are equivalent to [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] are:
[tex]\[ \boxed{C,\, E} \][/tex]
### Step 1: Understand the Original Polynomial
The polynomial we need to match is:
[tex]\[ 6x^4 + 4x^3 - 7x^2 + 5x + 8 \][/tex]
### Step 2: Analyze Each Choice
Choice A: [tex]\(16x^{10}\)[/tex]
- This is a single term polynomial and is clearly not in the form of the original polynomial, which has multiple terms.
- Therefore, [tex]\(16x^{10}\)[/tex] is not equivalent to the original polynomial.
Choice B: [tex]\(6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x\)[/tex]
- Compare the terms of this polynomial to the original:
- The degree of the leading term here is [tex]\(x^5\)[/tex], whereas our original polynomial has a leading term of [tex]\(x^4\)[/tex].
- Additionally, the powers and coefficients of the subsequent terms do not match the original polynomial.
- Therefore, [tex]\(6x^5 + 4x^4 - 7x^3 + 5x^2 + 8x\)[/tex] is not equivalent to the original polynomial.
Choice C: [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex]
- This exactly matches our original polynomial term for term.
- Therefore, [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] is equivalent to the original polynomial.
Choice D: [tex]\(8 + 5x + 7x^2 - 4x^3 + 6x^4\)[/tex]
- Let's reorder this polynomial to compare more easily with the original:
- Ordered polynomial is [tex]\(6x^4 - 4x^3 + 7x^2 + 5x + 8\)[/tex].
- Although the independent term and linear term are the same, the signs and coefficients of other terms do not match.
- Therefore, [tex]\(8 + 5x + 7x^2 - 4x^3 + 6x^4\)[/tex] is not equivalent to the original polynomial.
Choice E: [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex]
- Let's reorder this polynomial to compare more easily with the original:
- Ordered polynomial is [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex].
- This exactly matches our original polynomial term for term.
- Therefore, [tex]\(8 + 5x - 7x^2 + 4x^3 + 6x^4\)[/tex] is equivalent to the original polynomial.
### Step 3: Conclusion
Based on our detailed comparisons, the polynomial expressions that are equivalent to [tex]\(6x^4 + 4x^3 - 7x^2 + 5x + 8\)[/tex] are:
[tex]\[ \boxed{C,\, E} \][/tex]
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