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

[tex]3x^3 + 9x^7 - x + 4x^{12}[/tex]

A. [tex]3x^3 + 4x^{12} + 9x^7 - x[/tex]

B. [tex]4x^{12} + 3x^3 - x + 9x^7[/tex]

C. [tex]4x^{12} + 9x^7 + 3x^3 - x[/tex]

D. [tex]9x^7 + 4x^{12} + 3x^3 - x[/tex]

To write the polynomial [tex]\(3x^3 + 9x^7 - x + 4x^{12}\)[/tex] in descending order, we need to sort the terms by the power of [tex]\(x\)[/tex] starting from the highest exponent to the lowest.

Let's break it down step-by-step:

1. Identify the terms and their exponents:
- [tex]\(3x^3\)[/tex] has an exponent of 3.
- [tex]\(9x^7\)[/tex] has an exponent of 7.
- [tex]\(-x\)[/tex] can be considered as [tex]\(-1x^1\)[/tex], with an exponent of 1.
- [tex]\(4x^{12}\)[/tex] has an exponent of 12.

2. Order the terms by exponent from highest to lowest:
- The term with the highest exponent is [tex]\(4x^{12}\)[/tex].
- The next highest exponent is 7, which is [tex]\(9x^7\)[/tex].
- Next, we have [tex]\(3x^3\)[/tex] with an exponent of 3.
- Finally, [tex]\(-x\)[/tex] (or [tex]\(-1x^1\)[/tex]) has the lowest exponent of 1.

3. Rearrange the terms in descending order based on their exponents:
- Start with the term with the exponent 12: [tex]\(4x^{12}\)[/tex].
- Then the term with the exponent 7: [tex]\(9x^7\)[/tex].
- Followed by the term with the exponent 3: [tex]\(3x^3\)[/tex].
- End with the term with the exponent 1: [tex]\(-x\)[/tex].

So, the polynomial written in descending order is:
[tex]\[4x^{12} + 9x^7 + 3x^3 - x\][/tex]

This matches option C. Therefore, the correct answer is:
Option C: [tex]\(4x^{12} + 9x^7 + 3x^3 - x\)[/tex]