Answer :
To solve the problem of arranging the polynomial [tex]\(4x^2 - x + 8x^6 + 3 + 2x^{10}\)[/tex] in descending order, follow these steps:
1. Identify the Powers of [tex]\(x\)[/tex]:
- Look at each term and note the exponent on [tex]\(x\)[/tex].
- The terms are:
- [tex]\(4x^2\)[/tex] (where the exponent is 2)
- [tex]\(-x\)[/tex] (which is [tex]\(-1\)[/tex] and can be written as [tex]\(-1x^1\)[/tex], where the exponent is 1)
- [tex]\(8x^6\)[/tex] (where the exponent is 6)
- [tex]\(3\)[/tex] (which is a constant, effectively [tex]\(x^0\)[/tex])
- [tex]\(2x^{10}\)[/tex] (where the exponent is 10)
2. Order the Terms by Exponent:
- List the terms by decreasing exponent values:
- The highest exponent is 10: [tex]\(2x^{10}\)[/tex]
- Next is 6: [tex]\(8x^6\)[/tex]
- Then 2: [tex]\(4x^2\)[/tex]
- Then 1: [tex]\(-x\)[/tex]
- The constant term (exponent 0) is mentioned last: [tex]\(+ 3\)[/tex]
3. Combine the Ordered Terms:
- The polynomial in descending order is:
[tex]\[
2x^{10} + 8x^6 + 4x^2 - x + 3
\][/tex]
4. Match With Given Options:
- Compare this ordered expression with the options given:
- Option A: [tex]\(2x^{10} + 8x^5 + 4x^2 - x + 3\)[/tex]
- Option B: [tex]\(8x^6 + 4x^2 + 3 + 2x^{10} - x\)[/tex]
- Option C: [tex]\(2x^{10} + 4x^2 - x + 3 + 8x^6\)[/tex]
- Option D: [tex]\(3 + 2x^{10} + 8x^6 + 4x^2 - x\)[/tex]
The polynomial ordered in descending powers of [tex]\(x\)[/tex] matches Option D.
Thus, the polynomial correctly written in descending order is represented in Option D: [tex]\(3 + 2x^{10} + 8x^6 + 4x^2 - x\)[/tex].
1. Identify the Powers of [tex]\(x\)[/tex]:
- Look at each term and note the exponent on [tex]\(x\)[/tex].
- The terms are:
- [tex]\(4x^2\)[/tex] (where the exponent is 2)
- [tex]\(-x\)[/tex] (which is [tex]\(-1\)[/tex] and can be written as [tex]\(-1x^1\)[/tex], where the exponent is 1)
- [tex]\(8x^6\)[/tex] (where the exponent is 6)
- [tex]\(3\)[/tex] (which is a constant, effectively [tex]\(x^0\)[/tex])
- [tex]\(2x^{10}\)[/tex] (where the exponent is 10)
2. Order the Terms by Exponent:
- List the terms by decreasing exponent values:
- The highest exponent is 10: [tex]\(2x^{10}\)[/tex]
- Next is 6: [tex]\(8x^6\)[/tex]
- Then 2: [tex]\(4x^2\)[/tex]
- Then 1: [tex]\(-x\)[/tex]
- The constant term (exponent 0) is mentioned last: [tex]\(+ 3\)[/tex]
3. Combine the Ordered Terms:
- The polynomial in descending order is:
[tex]\[
2x^{10} + 8x^6 + 4x^2 - x + 3
\][/tex]
4. Match With Given Options:
- Compare this ordered expression with the options given:
- Option A: [tex]\(2x^{10} + 8x^5 + 4x^2 - x + 3\)[/tex]
- Option B: [tex]\(8x^6 + 4x^2 + 3 + 2x^{10} - x\)[/tex]
- Option C: [tex]\(2x^{10} + 4x^2 - x + 3 + 8x^6\)[/tex]
- Option D: [tex]\(3 + 2x^{10} + 8x^6 + 4x^2 - x\)[/tex]
The polynomial ordered in descending powers of [tex]\(x\)[/tex] matches Option D.
Thus, the polynomial correctly written in descending order is represented in Option D: [tex]\(3 + 2x^{10} + 8x^6 + 4x^2 - x\)[/tex].
