Answer :
We need to determine which polynomial has its terms arranged so that the exponents are in descending order. Let's analyze each option by listing the exponents of [tex]$x$[/tex] in the order they appear:
1. In Option A, the polynomial is
[tex]$$10x^2 + 8x^3 + x^8 - 2 + 3x^6.$$[/tex]
The exponents here are [tex]$2$[/tex], [tex]$3$[/tex], [tex]$8$[/tex], [tex]$0$[/tex] (from the constant [tex]$-2$[/tex]), and [tex]$6$[/tex]. This gives the sequence:
[tex]$$2,\,3,\,8,\,0,\,6.$$[/tex]
This sequence is not in strictly descending order.
2. In Option B, the polynomial is
[tex]$$x^8 + 10x^2 + 8x^3 + 3x^6 - 2.$$[/tex]
The exponents here are [tex]$8$[/tex], [tex]$2$[/tex], [tex]$3$[/tex], [tex]$6$[/tex], and [tex]$0$[/tex]. This sequence is:
[tex]$$8,\,2,\,3,\,6,\,0.$$[/tex]
This sequence is also not in descending order.
3. In Option C, the polynomial is
[tex]$$x^8 + 3x^6 + 8x^3 + 10x^2 - 2.$$[/tex]
The exponents here are [tex]$8$[/tex], [tex]$6$[/tex], [tex]$3$[/tex], [tex]$2$[/tex], and [tex]$0$[/tex]. This gives:
[tex]$$8,\,6,\,3,\,2,\,0.$$[/tex]
The sequence is in descending order, as each term is smaller than or equal to the preceding one.
4. In Option D, the polynomial is
[tex]$$3x^6 + 10x^2 + x^8 + 8x^3 - 2.$$[/tex]
The exponents here are [tex]$6$[/tex], [tex]$2$[/tex], [tex]$8$[/tex], [tex]$3$[/tex], and [tex]$0$[/tex]. This sequence is:
[tex]$$6,\,2,\,8,\,3,\,0.$$[/tex]
This is not in descending order.
Since Option C is the only one where the exponents appear in descending order, it is the correct choice.
The final answer is: Option C.
1. In Option A, the polynomial is
[tex]$$10x^2 + 8x^3 + x^8 - 2 + 3x^6.$$[/tex]
The exponents here are [tex]$2$[/tex], [tex]$3$[/tex], [tex]$8$[/tex], [tex]$0$[/tex] (from the constant [tex]$-2$[/tex]), and [tex]$6$[/tex]. This gives the sequence:
[tex]$$2,\,3,\,8,\,0,\,6.$$[/tex]
This sequence is not in strictly descending order.
2. In Option B, the polynomial is
[tex]$$x^8 + 10x^2 + 8x^3 + 3x^6 - 2.$$[/tex]
The exponents here are [tex]$8$[/tex], [tex]$2$[/tex], [tex]$3$[/tex], [tex]$6$[/tex], and [tex]$0$[/tex]. This sequence is:
[tex]$$8,\,2,\,3,\,6,\,0.$$[/tex]
This sequence is also not in descending order.
3. In Option C, the polynomial is
[tex]$$x^8 + 3x^6 + 8x^3 + 10x^2 - 2.$$[/tex]
The exponents here are [tex]$8$[/tex], [tex]$6$[/tex], [tex]$3$[/tex], [tex]$2$[/tex], and [tex]$0$[/tex]. This gives:
[tex]$$8,\,6,\,3,\,2,\,0.$$[/tex]
The sequence is in descending order, as each term is smaller than or equal to the preceding one.
4. In Option D, the polynomial is
[tex]$$3x^6 + 10x^2 + x^8 + 8x^3 - 2.$$[/tex]
The exponents here are [tex]$6$[/tex], [tex]$2$[/tex], [tex]$8$[/tex], [tex]$3$[/tex], and [tex]$0$[/tex]. This sequence is:
[tex]$$6,\,2,\,8,\,3,\,0.$$[/tex]
This is not in descending order.
Since Option C is the only one where the exponents appear in descending order, it is the correct choice.
The final answer is: Option C.
