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------------------------------------------------ Which of the following shows the polynomial below written in descending order?

[tex] 4x^2 - x + 8x^6 + 3 + 2x^{10} [/tex]

A. [tex] 8x^6 + 4x^2 + 3 + 2x^{10} - x [/tex]

B. [tex] 2x^{10} + 4x^2 - x + 3 + 8x^6 [/tex]

C. [tex] 2x^{10} + 8x^6 + 4x^2 - x + 3 [/tex]

D. [tex] 3 + 2x^{10} + 8x^6 + 4x^2 - x [/tex]

To write the polynomial [tex]$4x^2 - x + 8x^6 + 3 + 2x^{10}$[/tex] in descending order, you must organize the terms from the highest exponent to the lowest exponent. Here’s how you can do it:

1. Identify the exponents: Look at each term and its exponent:
- [tex]$2x^{10}$[/tex] has an exponent of 10.
- [tex]$8x^6$[/tex] has an exponent of 6.
- [tex]$4x^2$[/tex] has an exponent of 2.
- [tex]$-x$[/tex] (which is the same as [tex]$-1x^1$[/tex]) has an exponent of 1.
- [tex]$3$[/tex] doesn't have a variable, so consider it as [tex]$3x^0$[/tex] with an exponent of 0.

2. Order the terms by exponents: Arrange the terms from the highest exponent to the lowest:
- The highest exponent is 10, so the first term is [tex]$2x^{10}$[/tex].
- The next highest exponent is 6, so the next term is [tex]$8x^6$[/tex].
- The next highest exponent is 2, so [tex]$4x^2$[/tex] comes next.
- Followed by [tex]$-x$[/tex] which has an exponent of 1.
- Finally, [tex]$3$[/tex] with an exponent of 0.

3. Write the polynomial: After arranging the terms, the polynomial written in descending order is:
[tex]\[
2x^{10} + 8x^6 + 4x^2 - x + 3
\][/tex]

Looking at the given options, option C matches this arrangement:
- C. [tex]$2x^{10} + 8x^6 + 4x^2 - x + 3$[/tex]

Therefore, the correct answer is C.