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Answer :
To add the polynomials [tex]\((8x^8 - 9x^3 + 3x^2 + 9)\)[/tex] and [tex]\((4x^7 + 6x^3 - 2x)\)[/tex], we will combine like terms. Here's how to do it step-by-step:
1. Identify like terms in the polynomials:
- The first polynomial is [tex]\(8x^8 - 9x^3 + 3x^2 + 9\)[/tex].
- The second polynomial is [tex]\(4x^7 + 6x^3 - 2x\)[/tex].
2. Combine the coefficients of like terms:
- For the term [tex]\(x^8\)[/tex], we only have [tex]\(8x^8\)[/tex] from the first polynomial.
- For the term [tex]\(x^7\)[/tex], we have [tex]\(4x^7\)[/tex] from the second polynomial.
- For the term [tex]\(x^3\)[/tex], add: [tex]\(-9x^3\)[/tex] (from the first polynomial) and [tex]\(6x^3\)[/tex] (from the second polynomial), giving us [tex]\((-9 + 6)x^3 = -3x^3\)[/tex].
- For the term [tex]\(x^2\)[/tex], we only have [tex]\(3x^2\)[/tex] from the first polynomial.
- For the term [tex]\(x\)[/tex], we have [tex]\(-2x\)[/tex] from the second polynomial.
- The constant term is [tex]\(9\)[/tex] from the first polynomial.
3. Construct the resulting polynomial:
- Combine all the terms:
[tex]\[
8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9
\][/tex]
This gives us the final polynomial:
[tex]\[
8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9
\][/tex]
Thus, the correct answer is option D: [tex]\(8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9\)[/tex].
1. Identify like terms in the polynomials:
- The first polynomial is [tex]\(8x^8 - 9x^3 + 3x^2 + 9\)[/tex].
- The second polynomial is [tex]\(4x^7 + 6x^3 - 2x\)[/tex].
2. Combine the coefficients of like terms:
- For the term [tex]\(x^8\)[/tex], we only have [tex]\(8x^8\)[/tex] from the first polynomial.
- For the term [tex]\(x^7\)[/tex], we have [tex]\(4x^7\)[/tex] from the second polynomial.
- For the term [tex]\(x^3\)[/tex], add: [tex]\(-9x^3\)[/tex] (from the first polynomial) and [tex]\(6x^3\)[/tex] (from the second polynomial), giving us [tex]\((-9 + 6)x^3 = -3x^3\)[/tex].
- For the term [tex]\(x^2\)[/tex], we only have [tex]\(3x^2\)[/tex] from the first polynomial.
- For the term [tex]\(x\)[/tex], we have [tex]\(-2x\)[/tex] from the second polynomial.
- The constant term is [tex]\(9\)[/tex] from the first polynomial.
3. Construct the resulting polynomial:
- Combine all the terms:
[tex]\[
8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9
\][/tex]
This gives us the final polynomial:
[tex]\[
8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9
\][/tex]
Thus, the correct answer is option D: [tex]\(8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9\)[/tex].
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