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Answer :
To find the sum of the polynomials [tex]\((2x^4 + 5x^3 - 8x^2 - x + 10)\)[/tex] and [tex]\((8x^4 - 4x^3 + x^2 - x + 2)\)[/tex], we simply add the corresponding coefficients from each polynomial. Let's break it down step-by-step:
1. Identify the coefficients:
- For [tex]\(x^4\)[/tex], we have [tex]\(2\)[/tex] from the first polynomial and [tex]\(8\)[/tex] from the second.
- For [tex]\(x^3\)[/tex], we have [tex]\(5\)[/tex] from the first polynomial and [tex]\(-4\)[/tex] from the second.
- For [tex]\(x^2\)[/tex], we have [tex]\(-8\)[/tex] from the first polynomial and [tex]\(1\)[/tex] from the second.
- For [tex]\(x\)[/tex], we have [tex]\(-1\)[/tex] from both polynomials.
- For the constant term, we have [tex]\(10\)[/tex] from the first polynomial and [tex]\(2\)[/tex] from the second.
2. Add the coefficients:
- [tex]\(x^4\)[/tex] term: [tex]\(2 + 8 = 10\)[/tex]
- [tex]\(x^3\)[/tex] term: [tex]\(5 - 4 = 1\)[/tex]
- [tex]\(x^2\)[/tex] term: [tex]\(-8 + 1 = -7\)[/tex]
- [tex]\(x\)[/tex] term: [tex]\(-1 - 1 = -2\)[/tex]
- Constant term: [tex]\(10 + 2 = 12\)[/tex]
3. Combine the results into the polynomial:
[tex]\[
10x^4 + 1x^3 - 7x^2 - 2x + 12
\][/tex]
The sum of the polynomials is [tex]\(10x^4 + x^3 - 7x^2 - 2x + 12\)[/tex].
1. Identify the coefficients:
- For [tex]\(x^4\)[/tex], we have [tex]\(2\)[/tex] from the first polynomial and [tex]\(8\)[/tex] from the second.
- For [tex]\(x^3\)[/tex], we have [tex]\(5\)[/tex] from the first polynomial and [tex]\(-4\)[/tex] from the second.
- For [tex]\(x^2\)[/tex], we have [tex]\(-8\)[/tex] from the first polynomial and [tex]\(1\)[/tex] from the second.
- For [tex]\(x\)[/tex], we have [tex]\(-1\)[/tex] from both polynomials.
- For the constant term, we have [tex]\(10\)[/tex] from the first polynomial and [tex]\(2\)[/tex] from the second.
2. Add the coefficients:
- [tex]\(x^4\)[/tex] term: [tex]\(2 + 8 = 10\)[/tex]
- [tex]\(x^3\)[/tex] term: [tex]\(5 - 4 = 1\)[/tex]
- [tex]\(x^2\)[/tex] term: [tex]\(-8 + 1 = -7\)[/tex]
- [tex]\(x\)[/tex] term: [tex]\(-1 - 1 = -2\)[/tex]
- Constant term: [tex]\(10 + 2 = 12\)[/tex]
3. Combine the results into the polynomial:
[tex]\[
10x^4 + 1x^3 - 7x^2 - 2x + 12
\][/tex]
The sum of the polynomials is [tex]\(10x^4 + x^3 - 7x^2 - 2x + 12\)[/tex].
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