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Answer :
Sure, let's determine which polynomial lists the powers in descending order by examining each option step-by-step.
We need to arrange the terms by the exponent of [tex]\( x \)[/tex] from the highest to the lowest.
Option A: [tex]\( 10x^2 + 8x^3 + x^8 - 2 + 3x^6 \)[/tex]
- The terms in this polynomial are:
- [tex]\( x^8 \)[/tex] (power 8)
- [tex]\( 3x^6 \)[/tex] (power 6)
- [tex]\( 8x^3 \)[/tex] (power 3)
- [tex]\( 10x^2 \)[/tex] (power 2)
- [tex]\(-2\)[/tex] (constant term, power 0)
Rearranging terms:
[tex]\[ x^8 + 3x^6 + 8x^3 + 10x^2 - 2 \][/tex]
Option B: [tex]\( x^8 + 10x^2 + 8x^3 + 3x^6 - 2 \)[/tex]
- The terms in this polynomial are:
- [tex]\( x^8 \)[/tex] (power 8)
- [tex]\( 3x^6 \)[/tex] (power 6)
- [tex]\( 8x^3 \)[/tex] (power 3)
- [tex]\( 10x^2 \)[/tex] (power 2)
- [tex]\(-2\)[/tex] (constant term, power 0)
Current order: [tex]\( x^8 \)[/tex], [tex]\( 10x^2 \)[/tex], [tex]\( 8x^3 \)[/tex], [tex]\( 3x^6 \)[/tex], [tex]\(-2 \)[/tex] – Not in descending order.
Option C: [tex]\( x^8 + 3x^6 + 8x^3 + 10x^2 - 2 \)[/tex]
- The terms in this polynomial are:
- [tex]\( x^8 \)[/tex] (power 8)
- [tex]\( 3x^6 \)[/tex] (power 6)
- [tex]\( 8x^3 \)[/tex] (power 3)
- [tex]\( 10x^2 \)[/tex] (power 2)
- [tex]\(-2\)[/tex] (constant term, power 0)
Order already: [tex]\( x^8 \)[/tex], [tex]\( 3x^6 \)[/tex], [tex]\( 8x^3 \)[/tex], [tex]\( 10x^2 \)[/tex], [tex]\(-2 \)[/tex] – In descending order.
Option D: [tex]\( 3x^6 + 10x^2 + x^8 + 8x^3 - 2 \)[/tex]
- The terms in this polynomial are:
- [tex]\( x^8 \)[/tex] (power 8)
- [tex]\( 3x^6 \)[/tex] (power 6)
- [tex]\( 8x^3 \)[/tex] (power 3)
- [tex]\( 10x^2 \)[/tex] (power 2)
- [tex]\(-2\)[/tex] (constant term, power 0)
Current order: [tex]\( 3x^6 \)[/tex], [tex]\( 10x^2 \)[/tex], [tex]\( x^8 \)[/tex], [tex]\( 8x^3 \)[/tex], [tex]\(-2 \)[/tex] – Not in descending order.
So, after analyzing all the options, the polynomial that lists the powers in descending order is:
Option C: [tex]\( x^8 + 3x^6 + 8x^3 + 10x^2 - 2 \)[/tex]
This polynomial directly lists the terms in order from the highest exponent to the lowest without requiring rearrangement.
We need to arrange the terms by the exponent of [tex]\( x \)[/tex] from the highest to the lowest.
Option A: [tex]\( 10x^2 + 8x^3 + x^8 - 2 + 3x^6 \)[/tex]
- The terms in this polynomial are:
- [tex]\( x^8 \)[/tex] (power 8)
- [tex]\( 3x^6 \)[/tex] (power 6)
- [tex]\( 8x^3 \)[/tex] (power 3)
- [tex]\( 10x^2 \)[/tex] (power 2)
- [tex]\(-2\)[/tex] (constant term, power 0)
Rearranging terms:
[tex]\[ x^8 + 3x^6 + 8x^3 + 10x^2 - 2 \][/tex]
Option B: [tex]\( x^8 + 10x^2 + 8x^3 + 3x^6 - 2 \)[/tex]
- The terms in this polynomial are:
- [tex]\( x^8 \)[/tex] (power 8)
- [tex]\( 3x^6 \)[/tex] (power 6)
- [tex]\( 8x^3 \)[/tex] (power 3)
- [tex]\( 10x^2 \)[/tex] (power 2)
- [tex]\(-2\)[/tex] (constant term, power 0)
Current order: [tex]\( x^8 \)[/tex], [tex]\( 10x^2 \)[/tex], [tex]\( 8x^3 \)[/tex], [tex]\( 3x^6 \)[/tex], [tex]\(-2 \)[/tex] – Not in descending order.
Option C: [tex]\( x^8 + 3x^6 + 8x^3 + 10x^2 - 2 \)[/tex]
- The terms in this polynomial are:
- [tex]\( x^8 \)[/tex] (power 8)
- [tex]\( 3x^6 \)[/tex] (power 6)
- [tex]\( 8x^3 \)[/tex] (power 3)
- [tex]\( 10x^2 \)[/tex] (power 2)
- [tex]\(-2\)[/tex] (constant term, power 0)
Order already: [tex]\( x^8 \)[/tex], [tex]\( 3x^6 \)[/tex], [tex]\( 8x^3 \)[/tex], [tex]\( 10x^2 \)[/tex], [tex]\(-2 \)[/tex] – In descending order.
Option D: [tex]\( 3x^6 + 10x^2 + x^8 + 8x^3 - 2 \)[/tex]
- The terms in this polynomial are:
- [tex]\( x^8 \)[/tex] (power 8)
- [tex]\( 3x^6 \)[/tex] (power 6)
- [tex]\( 8x^3 \)[/tex] (power 3)
- [tex]\( 10x^2 \)[/tex] (power 2)
- [tex]\(-2\)[/tex] (constant term, power 0)
Current order: [tex]\( 3x^6 \)[/tex], [tex]\( 10x^2 \)[/tex], [tex]\( x^8 \)[/tex], [tex]\( 8x^3 \)[/tex], [tex]\(-2 \)[/tex] – Not in descending order.
So, after analyzing all the options, the polynomial that lists the powers in descending order is:
Option C: [tex]\( x^8 + 3x^6 + 8x^3 + 10x^2 - 2 \)[/tex]
This polynomial directly lists the terms in order from the highest exponent to the lowest without requiring rearrangement.
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