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

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

To write the polynomial [tex]\(3x^3 + 9x^7 - x + 4x^{12}\)[/tex] in descending order, we need to arrange the terms so that the powers of [tex]\(x\)[/tex] are in decreasing order. Here's how you can do it step by step:

1. Identify the exponents: Look at each term in the polynomial and note the exponent of [tex]\(x\)[/tex]:
- [tex]\(3x^3\)[/tex] has an exponent of 3.
- [tex]\(9x^7\)[/tex] has an exponent of 7.
- [tex]\(-x\)[/tex] can be rewritten as [tex]\(-1x^1\)[/tex], which has an exponent of 1.
- [tex]\(4x^{12}\)[/tex] has an exponent of 12.

2. Order the terms by exponents: Arrange the terms from the highest exponent to the lowest:
- The highest exponent is 12, so [tex]\(4x^{12}\)[/tex] comes first.
- Next is the exponent 7, which corresponds to [tex]\(9x^7\)[/tex].
- Then the exponent 3, which is [tex]\(3x^3\)[/tex].
- Finally, the exponent 1, which is [tex]\(-x\)[/tex].

3. Write the polynomial in descending order: Putting these terms together, maintaining their original coefficients and signs, we get:
- [tex]\(4x^{12} + 9x^7 + 3x^3 - x\)[/tex]

This matches choice D in the list of options.

Therefore, the polynomial written in descending order is [tex]\(4x^{12} + 9x^7 + 3x^3 - x\)[/tex], which corresponds to option D.