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Answer :
To write the polynomial [tex]\(3x^3 + 9x^7 - x + 4x^{12}\)[/tex] in descending order, we need to arrange the terms by the exponents of [tex]\(x\)[/tex] from highest to lowest. Here’s a step-by-step solution:
1. Identify the exponents in the polynomial:
- The term [tex]\(3x^3\)[/tex] has an exponent of 3.
- The term [tex]\(9x^7\)[/tex] has an exponent of 7.
- The term [tex]\(-x\)[/tex] can be written as [tex]\(-x^1\)[/tex] and has an exponent of 1.
- The term [tex]\(4x^{12}\)[/tex] has an exponent of 12.
2. List the exponents in numerical order for clarity:
- 12, 7, 3, 1
3. Arrange the terms based on these exponents from highest to lowest:
- The highest exponent is 12, so the first term is [tex]\(4x^{12}\)[/tex].
- The next highest exponent is 7, so the next term is [tex]\(9x^7\)[/tex].
- The next term with an exponent of 3 is [tex]\(3x^3\)[/tex].
- The term with the lowest exponent of 1 is [tex]\(-x\)[/tex].
4. Write the polynomial with the terms in descending order:
- [tex]\(4x^{12} + 9x^7 + 3x^3 - x\)[/tex]
Comparing this to the provided options:
- A. [tex]\(4x^{12} + 9x^7 + 3x^3 - x\)[/tex]
- B. [tex]\(9x^7 + 4x^{12} + 3x^3 - x\)[/tex]
- C. [tex]\(4x^{12} + 3x^3 - x + 9x^7\)[/tex]
- D. [tex]\(3x^3 + 4x^{12} + 9x^7 - x\)[/tex]
Option A is correct because it matches our polynomial in descending order.
Thus, the correct answer is:
A. [tex]\(4x^{12} + 9x^7 + 3x^3 - x\)[/tex]
1. Identify the exponents in the polynomial:
- The term [tex]\(3x^3\)[/tex] has an exponent of 3.
- The term [tex]\(9x^7\)[/tex] has an exponent of 7.
- The term [tex]\(-x\)[/tex] can be written as [tex]\(-x^1\)[/tex] and has an exponent of 1.
- The term [tex]\(4x^{12}\)[/tex] has an exponent of 12.
2. List the exponents in numerical order for clarity:
- 12, 7, 3, 1
3. Arrange the terms based on these exponents from highest to lowest:
- The highest exponent is 12, so the first term is [tex]\(4x^{12}\)[/tex].
- The next highest exponent is 7, so the next term is [tex]\(9x^7\)[/tex].
- The next term with an exponent of 3 is [tex]\(3x^3\)[/tex].
- The term with the lowest exponent of 1 is [tex]\(-x\)[/tex].
4. Write the polynomial with the terms in descending order:
- [tex]\(4x^{12} + 9x^7 + 3x^3 - x\)[/tex]
Comparing this to the provided options:
- A. [tex]\(4x^{12} + 9x^7 + 3x^3 - x\)[/tex]
- B. [tex]\(9x^7 + 4x^{12} + 3x^3 - x\)[/tex]
- C. [tex]\(4x^{12} + 3x^3 - x + 9x^7\)[/tex]
- D. [tex]\(3x^3 + 4x^{12} + 9x^7 - x\)[/tex]
Option A is correct because it matches our polynomial in descending order.
Thus, the correct answer is:
A. [tex]\(4x^{12} + 9x^7 + 3x^3 - x\)[/tex]
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