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Answer :
To write the given polynomial [tex]\(5x^3 - x + 9x^7 + 4 + 3x^{11}\)[/tex] in descending order, you need to arrange the terms by their exponents, starting with the highest exponent and moving to the lowest.
Here's how you can do it step-by-step:
1. Identify the Exponents: Look at each term in the polynomial and note the exponents of [tex]\(x\)[/tex]:
- [tex]\(3x^{11}\)[/tex] has an exponent of 11.
- [tex]\(9x^7\)[/tex] has an exponent of 7.
- [tex]\(5x^3\)[/tex] has an exponent of 3.
- [tex]\(-x\)[/tex] is the same as [tex]\(-1x^1\)[/tex], so it has an exponent of 1.
- [tex]\(4\)[/tex] is a constant term, which can be considered as [tex]\(4x^0\)[/tex], with an exponent of 0.
2. Order the Terms by Exponents: Arrange these terms in descending order based on the exponents:
- Start with the term with the highest exponent: [tex]\(3x^{11}\)[/tex].
- Next is the term with the next highest exponent: [tex]\(9x^7\)[/tex].
- Continue with the next: [tex]\(5x^3\)[/tex].
- Then the linear term: [tex]\(-x\)[/tex].
- Finally, the constant term: [tex]\(+4\)[/tex].
3. Write the Polynomial: Combine all the terms you've arranged:
[tex]\[
3x^{11} + 9x^7 + 5x^3 - x + 4
\][/tex]
Therefore, the polynomial written in descending order is:
[tex]\[ 3x^{11} + 9x^7 + 5x^3 - x + 4 \][/tex]
This corresponds to option B from the choices provided:
[tex]\[ \text{B. } 3x^{11} + 9x^7 + 5x^3 - x + 4 \][/tex]
Here's how you can do it step-by-step:
1. Identify the Exponents: Look at each term in the polynomial and note the exponents of [tex]\(x\)[/tex]:
- [tex]\(3x^{11}\)[/tex] has an exponent of 11.
- [tex]\(9x^7\)[/tex] has an exponent of 7.
- [tex]\(5x^3\)[/tex] has an exponent of 3.
- [tex]\(-x\)[/tex] is the same as [tex]\(-1x^1\)[/tex], so it has an exponent of 1.
- [tex]\(4\)[/tex] is a constant term, which can be considered as [tex]\(4x^0\)[/tex], with an exponent of 0.
2. Order the Terms by Exponents: Arrange these terms in descending order based on the exponents:
- Start with the term with the highest exponent: [tex]\(3x^{11}\)[/tex].
- Next is the term with the next highest exponent: [tex]\(9x^7\)[/tex].
- Continue with the next: [tex]\(5x^3\)[/tex].
- Then the linear term: [tex]\(-x\)[/tex].
- Finally, the constant term: [tex]\(+4\)[/tex].
3. Write the Polynomial: Combine all the terms you've arranged:
[tex]\[
3x^{11} + 9x^7 + 5x^3 - x + 4
\][/tex]
Therefore, the polynomial written in descending order is:
[tex]\[ 3x^{11} + 9x^7 + 5x^3 - x + 4 \][/tex]
This corresponds to option B from the choices provided:
[tex]\[ \text{B. } 3x^{11} + 9x^7 + 5x^3 - x + 4 \][/tex]
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