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

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

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

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

To rewrite the polynomial [tex]\(3x^3 + 9x^7 - x + 4x^{12}\)[/tex] in descending order, you should arrange the terms from the highest degree to the lowest degree. Here's how to do it step-by-step:

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

2. Order the Exponents: Arrange these exponents in descending order: 12, 7, 3, 1.

3. Reorder the Terms: Write the terms of the polynomial according to the descending order of exponents:
- The term with the highest exponent is [tex]\(4x^{12}\)[/tex].
- Next is [tex]\(9x^7\)[/tex].
- Then comes [tex]\(3x^3\)[/tex].
- Finally, the term [tex]\( -x \)[/tex] (or [tex]\(-1x^1\)[/tex]).

4. Final Polynomial: Combine the reordered terms to get the polynomial in descending order:
[tex]\[
4x^{12} + 9x^7 + 3x^3 - x
\][/tex]

Now, compare this reordered polynomial with the given options:

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

The correct choice is A. [tex]\(4x^{12} + 9x^7 + 3x^3 - x\)[/tex] which matches the polynomial arranged in descending order.