Answer :
To determine which polynomial lists the powers in descending order, we need to arrange the terms of each polynomial by the exponents of [tex]\(x\)[/tex] from the highest to the lowest.
Let's analyze each option:
A. [tex]\(3x^6 + 10x^2 + x^8 + 8x^3 - 2\)[/tex]
- The exponents are 6, 2, 8, 3, and 0.
- If we rearrange these terms by decreasing powers: [tex]\(x^8, x^6, x^3, x^2, \text{and then the constant term}\)[/tex].
B. [tex]\(x^8 + 3x^6 + 8x^3 + 10x^2 - 2\)[/tex]
- The exponents are 8, 6, 3, 2, and 0.
- This polynomial already lists the terms in descending order: [tex]\(x^8, x^6, x^3, x^2, \text{and then the constant term}\)[/tex].
C. [tex]\(10x^2 + 8x^3 + x^8 - 2 + 3x^6\)[/tex]
- The exponents are 2, 3, 8, 0, and 6.
- If we rearrange these terms by decreasing powers: [tex]\(x^8, x^6, x^3, x^2, \text{and then the constant term}\)[/tex].
D. [tex]\(x^8 + 10x^2 + 8x^3 + 3x^6 - 2\)[/tex]
- The exponents are 8, 2, 3, 6, and 0.
- If we rearrange these terms by decreasing powers: [tex]\(x^8, x^6, x^3, x^2, \text{and then the constant term}\)[/tex].
From this analysis, the correct option that already arranges the terms in descending order by powers of [tex]\(x\)[/tex] is:
Option B: [tex]\(x^8 + 3x^6 + 8x^3 + 10x^2 - 2\)[/tex]
Let's analyze each option:
A. [tex]\(3x^6 + 10x^2 + x^8 + 8x^3 - 2\)[/tex]
- The exponents are 6, 2, 8, 3, and 0.
- If we rearrange these terms by decreasing powers: [tex]\(x^8, x^6, x^3, x^2, \text{and then the constant term}\)[/tex].
B. [tex]\(x^8 + 3x^6 + 8x^3 + 10x^2 - 2\)[/tex]
- The exponents are 8, 6, 3, 2, and 0.
- This polynomial already lists the terms in descending order: [tex]\(x^8, x^6, x^3, x^2, \text{and then the constant term}\)[/tex].
C. [tex]\(10x^2 + 8x^3 + x^8 - 2 + 3x^6\)[/tex]
- The exponents are 2, 3, 8, 0, and 6.
- If we rearrange these terms by decreasing powers: [tex]\(x^8, x^6, x^3, x^2, \text{and then the constant term}\)[/tex].
D. [tex]\(x^8 + 10x^2 + 8x^3 + 3x^6 - 2\)[/tex]
- The exponents are 8, 2, 3, 6, and 0.
- If we rearrange these terms by decreasing powers: [tex]\(x^8, x^6, x^3, x^2, \text{and then the constant term}\)[/tex].
From this analysis, the correct option that already arranges the terms in descending order by powers of [tex]\(x\)[/tex] is:
Option B: [tex]\(x^8 + 3x^6 + 8x^3 + 10x^2 - 2\)[/tex]
