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Answer :
To write the polynomial [tex]\(5x^3 - x + 9x^7 + 4 + 3x^{11}\)[/tex] in descending order, we need to arrange the terms starting from the highest power of [tex]\(x\)[/tex] to the lowest. Here’s how you can do it step-by-step:
1. Identify the Degrees of Each Term:
- [tex]\(5x^3\)[/tex] has a degree of 3.
- [tex]\(-x\)[/tex] (which is [tex]\(-1x^1\)[/tex]) has a degree of 1.
- [tex]\(9x^7\)[/tex] has a degree of 7.
- [tex]\(4\)[/tex] (a constant, i.e., [tex]\(4x^0\)[/tex]) has a degree of 0.
- [tex]\(3x^{11}\)[/tex] has a degree of 11.
2. Arrange the Terms in Descending Order of Degree:
- Start with the term with the highest degree: [tex]\(3x^{11}\)[/tex].
- Next is [tex]\(9x^7\)[/tex].
- Followed by [tex]\(5x^3\)[/tex].
- Then [tex]\(-x\)[/tex].
- Lastly, the constant term [tex]\(4\)[/tex].
3. Write the Polynomial in Descending Order:
- Arranging them by their degrees, we get: [tex]\(3x^{11} + 9x^7 + 5x^3 - x + 4\)[/tex].
4. Match with the Given Options:
We compare our sorted polynomial with the choices given:
- Option A: [tex]\(3x^{11} + 9x^7 - x + 4 + 5x^3\)[/tex]
- Option B: [tex]\(9x^7 + 5x^3 + 4 + 3x^{11} - x\)[/tex]
- Option C: [tex]\(4 + 3x^{11} + 9x^7 + 5x^3 - x\)[/tex]
- Option D: [tex]\(3x^{11} + 9x^7 + 5x^3 - x + 4\)[/tex]
The polynomial in descending order is matched exactly with Option D: [tex]\(3x^{11} + 9x^7 + 5x^3 - x + 4\)[/tex].
Thus, the correct answer is Option D.
1. Identify the Degrees of Each Term:
- [tex]\(5x^3\)[/tex] has a degree of 3.
- [tex]\(-x\)[/tex] (which is [tex]\(-1x^1\)[/tex]) has a degree of 1.
- [tex]\(9x^7\)[/tex] has a degree of 7.
- [tex]\(4\)[/tex] (a constant, i.e., [tex]\(4x^0\)[/tex]) has a degree of 0.
- [tex]\(3x^{11}\)[/tex] has a degree of 11.
2. Arrange the Terms in Descending Order of Degree:
- Start with the term with the highest degree: [tex]\(3x^{11}\)[/tex].
- Next is [tex]\(9x^7\)[/tex].
- Followed by [tex]\(5x^3\)[/tex].
- Then [tex]\(-x\)[/tex].
- Lastly, the constant term [tex]\(4\)[/tex].
3. Write the Polynomial in Descending Order:
- Arranging them by their degrees, we get: [tex]\(3x^{11} + 9x^7 + 5x^3 - x + 4\)[/tex].
4. Match with the Given Options:
We compare our sorted polynomial with the choices given:
- Option A: [tex]\(3x^{11} + 9x^7 - x + 4 + 5x^3\)[/tex]
- Option B: [tex]\(9x^7 + 5x^3 + 4 + 3x^{11} - x\)[/tex]
- Option C: [tex]\(4 + 3x^{11} + 9x^7 + 5x^3 - x\)[/tex]
- Option D: [tex]\(3x^{11} + 9x^7 + 5x^3 - x + 4\)[/tex]
The polynomial in descending order is matched exactly with Option D: [tex]\(3x^{11} + 9x^7 + 5x^3 - x + 4\)[/tex].
Thus, the correct answer is Option D.
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