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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 from the highest exponent of [tex]\(x\)[/tex] to the lowest.
Here's how to do it step-by-step:
1. Identify the exponents of the terms:
- The term [tex]\(5x^3\)[/tex] has an exponent of 3.
- The term [tex]\(-x\)[/tex] is equivalent to [tex]\(-1x^1\)[/tex], so it has an exponent of 1.
- The term [tex]\(9x^7\)[/tex] has an exponent of 7.
- The term [tex]\(4\)[/tex] is constant and can be thought of as having an exponent of 0.
- The term [tex]\(3x^{11}\)[/tex] has an exponent of 11.
2. Order the exponents from highest to lowest:
- The highest exponent is 11.
- Next is 7.
- Then 3.
- Then 1.
- Finally, the constant term 4 (exponent 0) comes last.
3. Arrange the terms according to the ordered exponents:
- Start with [tex]\(3x^{11}\)[/tex].
- Follow with [tex]\(9x^7\)[/tex].
- Then [tex]\(5x^3\)[/tex].
- Follow with [tex]\(-x\)[/tex].
- And lastly, the constant [tex]\(4\)[/tex].
So, the polynomial in descending order by exponents is:
[tex]\[ 3x^{11} + 9x^7 + 5x^3 - x + 4 \][/tex]
The option that matches this arrangement is Option C: [tex]\(3x^{11} + 9x^7 + 5x^3 - x + 4\)[/tex].
Here's how to do it step-by-step:
1. Identify the exponents of the terms:
- The term [tex]\(5x^3\)[/tex] has an exponent of 3.
- The term [tex]\(-x\)[/tex] is equivalent to [tex]\(-1x^1\)[/tex], so it has an exponent of 1.
- The term [tex]\(9x^7\)[/tex] has an exponent of 7.
- The term [tex]\(4\)[/tex] is constant and can be thought of as having an exponent of 0.
- The term [tex]\(3x^{11}\)[/tex] has an exponent of 11.
2. Order the exponents from highest to lowest:
- The highest exponent is 11.
- Next is 7.
- Then 3.
- Then 1.
- Finally, the constant term 4 (exponent 0) comes last.
3. Arrange the terms according to the ordered exponents:
- Start with [tex]\(3x^{11}\)[/tex].
- Follow with [tex]\(9x^7\)[/tex].
- Then [tex]\(5x^3\)[/tex].
- Follow with [tex]\(-x\)[/tex].
- And lastly, the constant [tex]\(4\)[/tex].
So, the polynomial in descending order by exponents is:
[tex]\[ 3x^{11} + 9x^7 + 5x^3 - x + 4 \][/tex]
The option that matches this arrangement is Option C: [tex]\(3x^{11} + 9x^7 + 5x^3 - x + 4\)[/tex].
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