Answer :
To determine which polynomial lists the powers in descending order, let's arrange the terms by looking at the exponents of [tex]\(x\)[/tex] in each option. We want to start with the largest power and progress to the smallest.
Here are the steps to solve the problem:
1. List the powers of [tex]\( x \)[/tex] in each option, starting from the highest power to the lowest:
- Option A: [tex]\( x^8 + 10x^2 + 8x^3 + 3x^6 - 2 \)[/tex]
- Arrangement: [tex]\(x^8, x^6, x^3, x^2, x^0\)[/tex]
- Option B: [tex]\( x^8 + 3x^6 + 8x^3 + 10x^2 - 2 \)[/tex]
- Arrangement: [tex]\(x^8, x^6, x^3, x^2, x^0\)[/tex]
- Option C: [tex]\( 3x^6 + 10x^2 + x^8 + 8x^3 - 2 \)[/tex]
- Arrangement: [tex]\(x^8, x^6, x^3, x^2, x^0\)[/tex]
- Option D: [tex]\( 10x^2 + 8x^3 + x^8 - 2 + 3x^6 \)[/tex]
- Arrangement: [tex]\(x^8, x^6, x^3, x^2, x^0\)[/tex]
2. Evaluate each option to see if they are in descending order:
- Analyzing Option A: The powers are not in order because after [tex]\( x^8 \)[/tex], the next term is [tex]\( 10x^2 \)[/tex], which should actually be [tex]\( x^6 \)[/tex].
- Analyzing Option B: This option correctly lists the powers in descending order: [tex]\( x^8, x^6, x^3, x^2, x^0 \)[/tex].
- Analyzing Option C: The highest power term [tex]\( x^8 \)[/tex] appears after [tex]\( 3x^6 \)[/tex] and [tex]\( 10x^2 \)[/tex], so it is not in descending order from the start.
- Analyzing Option D: The terms start with [tex]\( 10x^2 \)[/tex], and the highest power [tex]\( x^8 \)[/tex] appears later, which is incorrect for descending order.
3. Conclusion:
Option B: [tex]\( x^8 + 3x^6 + 8x^3 + 10x^2 - 2 \)[/tex] is the polynomial that lists the powers of [tex]\( x \)[/tex] in descending order correctly from highest to lowest.
Here are the steps to solve the problem:
1. List the powers of [tex]\( x \)[/tex] in each option, starting from the highest power to the lowest:
- Option A: [tex]\( x^8 + 10x^2 + 8x^3 + 3x^6 - 2 \)[/tex]
- Arrangement: [tex]\(x^8, x^6, x^3, x^2, x^0\)[/tex]
- Option B: [tex]\( x^8 + 3x^6 + 8x^3 + 10x^2 - 2 \)[/tex]
- Arrangement: [tex]\(x^8, x^6, x^3, x^2, x^0\)[/tex]
- Option C: [tex]\( 3x^6 + 10x^2 + x^8 + 8x^3 - 2 \)[/tex]
- Arrangement: [tex]\(x^8, x^6, x^3, x^2, x^0\)[/tex]
- Option D: [tex]\( 10x^2 + 8x^3 + x^8 - 2 + 3x^6 \)[/tex]
- Arrangement: [tex]\(x^8, x^6, x^3, x^2, x^0\)[/tex]
2. Evaluate each option to see if they are in descending order:
- Analyzing Option A: The powers are not in order because after [tex]\( x^8 \)[/tex], the next term is [tex]\( 10x^2 \)[/tex], which should actually be [tex]\( x^6 \)[/tex].
- Analyzing Option B: This option correctly lists the powers in descending order: [tex]\( x^8, x^6, x^3, x^2, x^0 \)[/tex].
- Analyzing Option C: The highest power term [tex]\( x^8 \)[/tex] appears after [tex]\( 3x^6 \)[/tex] and [tex]\( 10x^2 \)[/tex], so it is not in descending order from the start.
- Analyzing Option D: The terms start with [tex]\( 10x^2 \)[/tex], and the highest power [tex]\( x^8 \)[/tex] appears later, which is incorrect for descending order.
3. Conclusion:
Option B: [tex]\( x^8 + 3x^6 + 8x^3 + 10x^2 - 2 \)[/tex] is the polynomial that lists the powers of [tex]\( x \)[/tex] in descending order correctly from highest to lowest.
