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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 organize the terms by decreasing powers of [tex]\(x\)[/tex]. Here’s how to do it step-by-step:
1. Identify the powers of [tex]\(x\)[/tex] in each term:
- [tex]\(5x^3\)[/tex] has a power of 3.
- [tex]\(-x\)[/tex] (which is the same as [tex]\(-1x^1\)[/tex]) has a power of 1.
- [tex]\(9x^7\)[/tex] has a power of 7.
- [tex]\(4\)[/tex] is a constant, which can be considered as [tex]\(4x^0\)[/tex] with a power of 0.
- [tex]\(3x^{11}\)[/tex] has the highest power of 11.
2. List the terms in order of their powers, from highest to lowest:
- Start with the term that has the highest power of [tex]\(x\)[/tex], which is [tex]\(3x^{11}\)[/tex].
- Next is the term with the next highest power, [tex]\(9x^7\)[/tex].
- Then, the term [tex]\(5x^3\)[/tex].
- Followed by the term with [tex]\(x^1\)[/tex], which is [tex]\(-x\)[/tex].
- Lastly, the constant term [tex]\(4\)[/tex].
3. Write the polynomial in descending order:
- Assemble the terms in the order identified:
[tex]\[3x^{11} + 9x^7 + 5x^3 - x + 4\][/tex]
This arrangement is now in descending order of the powers of [tex]\(x\)[/tex]. Therefore, the polynomial written in descending order is:
[tex]\[3x^{11} + 9x^7 + 5x^3 - x + 4\][/tex]
1. Identify the powers of [tex]\(x\)[/tex] in each term:
- [tex]\(5x^3\)[/tex] has a power of 3.
- [tex]\(-x\)[/tex] (which is the same as [tex]\(-1x^1\)[/tex]) has a power of 1.
- [tex]\(9x^7\)[/tex] has a power of 7.
- [tex]\(4\)[/tex] is a constant, which can be considered as [tex]\(4x^0\)[/tex] with a power of 0.
- [tex]\(3x^{11}\)[/tex] has the highest power of 11.
2. List the terms in order of their powers, from highest to lowest:
- Start with the term that has the highest power of [tex]\(x\)[/tex], which is [tex]\(3x^{11}\)[/tex].
- Next is the term with the next highest power, [tex]\(9x^7\)[/tex].
- Then, the term [tex]\(5x^3\)[/tex].
- Followed by the term with [tex]\(x^1\)[/tex], which is [tex]\(-x\)[/tex].
- Lastly, the constant term [tex]\(4\)[/tex].
3. Write the polynomial in descending order:
- Assemble the terms in the order identified:
[tex]\[3x^{11} + 9x^7 + 5x^3 - x + 4\][/tex]
This arrangement is now in descending order of the powers of [tex]\(x\)[/tex]. Therefore, the polynomial written in descending order is:
[tex]\[3x^{11} + 9x^7 + 5x^3 - x + 4\][/tex]
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