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Answer :
To solve the problem of adding the two polynomials [tex]\((8x^8 - 9x^3 + 3x^2 + 9) + (4x^7 + 6x^3 - 2x)\)[/tex], we’ll combine like terms. Let's go step-by-step:
1. Identify and List Terms:
- From the first polynomial:
- [tex]\(8x^8\)[/tex]
- [tex]\(-9x^3\)[/tex]
- [tex]\(3x^2\)[/tex]
- [tex]\(9\)[/tex] (constant term)
- From the second polynomial:
- [tex]\(4x^7\)[/tex]
- [tex]\(6x^3\)[/tex]
- [tex]\(-2x\)[/tex]
2. Combine Like Terms:
- Only [tex]\(x^8\)[/tex] appears in the first polynomial, so the term [tex]\(8x^8\)[/tex] stays as is.
- For [tex]\(x^7\)[/tex], there is only [tex]\(4x^7\)[/tex] from the second polynomial.
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3\)[/tex] from the first polynomial and [tex]\(6x^3\)[/tex] from the second polynomial. The sum is [tex]\(-9x^3 + 6x^3 = -3x^3\)[/tex].
- The [tex]\(x^2\)[/tex] term, [tex]\(3x^2\)[/tex], only appears in the first polynomial.
- The [tex]\(x\)[/tex] term, [tex]\(-2x\)[/tex], only appears in the second polynomial.
- Lastly, the constant term is [tex]\(9\)[/tex].
3. Write the Resulting Polynomial:
- After combining like terms, the resulting polynomial is:
[tex]\[
8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9
\][/tex]
Therefore, the correct answer is Option D: [tex]\(8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9\)[/tex].
1. Identify and List Terms:
- From the first polynomial:
- [tex]\(8x^8\)[/tex]
- [tex]\(-9x^3\)[/tex]
- [tex]\(3x^2\)[/tex]
- [tex]\(9\)[/tex] (constant term)
- From the second polynomial:
- [tex]\(4x^7\)[/tex]
- [tex]\(6x^3\)[/tex]
- [tex]\(-2x\)[/tex]
2. Combine Like Terms:
- Only [tex]\(x^8\)[/tex] appears in the first polynomial, so the term [tex]\(8x^8\)[/tex] stays as is.
- For [tex]\(x^7\)[/tex], there is only [tex]\(4x^7\)[/tex] from the second polynomial.
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3\)[/tex] from the first polynomial and [tex]\(6x^3\)[/tex] from the second polynomial. The sum is [tex]\(-9x^3 + 6x^3 = -3x^3\)[/tex].
- The [tex]\(x^2\)[/tex] term, [tex]\(3x^2\)[/tex], only appears in the first polynomial.
- The [tex]\(x\)[/tex] term, [tex]\(-2x\)[/tex], only appears in the second polynomial.
- Lastly, the constant term is [tex]\(9\)[/tex].
3. Write the Resulting Polynomial:
- After combining like terms, the resulting polynomial is:
[tex]\[
8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9
\][/tex]
Therefore, the correct answer is Option D: [tex]\(8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9\)[/tex].
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