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Answer :
To determine which polynomial has its terms listed with exponents in strictly descending order, we need to look at the exponent of the variable in each term from left to right.
Let's examine each option:
1. For the polynomial
[tex]$$x^8+10x^2+8x^3+3x^6-2,$$[/tex]
the exponents are:
[tex]$$8, \;2, \;3, \;6, \;0.$$[/tex]
Notice that after the first term [tex]$x^8$[/tex], the exponents drop from 8 to 2, but then increase to 3 and 6. This sequence is not strictly descending.
2. For the polynomial
[tex]$$10x^2+8x^3+x^8-2+3x^6,$$[/tex]
the exponents are:
[tex]$$2, \;3, \;8, \;0, \;6.$$[/tex]
Here, the sequence is not in descending order because it starts with 2, increases to 3 and then to 8, among other discrepancies.
3. For the polynomial
[tex]$$x^8+3x^6+8x^3+10x^2-2,$$[/tex]
the exponents are:
[tex]$$8, \;6, \;3, \;2, \;0.$$[/tex]
This sequence is in strictly descending order since
[tex]$$8 > 6 > 3 > 2 > 0.$$[/tex]
4. For the polynomial
[tex]$$3x^6+10x^2+x^8+8x^3-2,$$[/tex]
the exponents are:
[tex]$$6, \;2, \;8, \;3, \;0.$$[/tex]
This is not in descending order since after 6 and 2, the exponent 8 appears, which is greater than 2.
Since only the third polynomial has its exponents arranged in descending order, the correct answer is option C.
Let's examine each option:
1. For the polynomial
[tex]$$x^8+10x^2+8x^3+3x^6-2,$$[/tex]
the exponents are:
[tex]$$8, \;2, \;3, \;6, \;0.$$[/tex]
Notice that after the first term [tex]$x^8$[/tex], the exponents drop from 8 to 2, but then increase to 3 and 6. This sequence is not strictly descending.
2. For the polynomial
[tex]$$10x^2+8x^3+x^8-2+3x^6,$$[/tex]
the exponents are:
[tex]$$2, \;3, \;8, \;0, \;6.$$[/tex]
Here, the sequence is not in descending order because it starts with 2, increases to 3 and then to 8, among other discrepancies.
3. For the polynomial
[tex]$$x^8+3x^6+8x^3+10x^2-2,$$[/tex]
the exponents are:
[tex]$$8, \;6, \;3, \;2, \;0.$$[/tex]
This sequence is in strictly descending order since
[tex]$$8 > 6 > 3 > 2 > 0.$$[/tex]
4. For the polynomial
[tex]$$3x^6+10x^2+x^8+8x^3-2,$$[/tex]
the exponents are:
[tex]$$6, \;2, \;8, \;3, \;0.$$[/tex]
This is not in descending order since after 6 and 2, the exponent 8 appears, which is greater than 2.
Since only the third polynomial has its exponents arranged in descending order, the correct answer is option C.
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