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

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

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

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

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

1. Identify the degree of each term:
- [tex]\(4x^{12}\)[/tex] has a degree of 12.
- [tex]\(9x^7\)[/tex] has a degree of 7.
- [tex]\(3x^3\)[/tex] has a degree of 3.
- [tex]\(-x\)[/tex] is the same as [tex]\(-x^1\)[/tex], so it has a degree of 1.

2. Arrange the terms in order of decreasing degree:
- The highest degree term is [tex]\(4x^{12}\)[/tex].
- The next highest is [tex]\(9x^7\)[/tex].
- Then comes [tex]\(3x^3\)[/tex].
- Finally, the lowest degree term is [tex]\(-x\)[/tex].

3. Write the polynomial with these terms in order:
- Start with [tex]\(4x^{12}\)[/tex].
- Add [tex]\(9x^7\)[/tex].
- Next, add [tex]\(3x^3\)[/tex].
- Finally, add [tex]\(-x\)[/tex].

The polynomial in descending order is:

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

This corresponds to option B:

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

Thus, the correct answer is option B.