Answer :
To add the two polynomials [tex]\((8x^8 - 9x^3 + 3x^2 + 9)\)[/tex] and [tex]\((4x^7 + 6x^3 - 2x)\)[/tex], we'll follow these steps:
1. Write the Polynomials Clearly:
- 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. Align Like Terms:
- Look for terms that have the same power of [tex]\(x\)[/tex]. These can be combined.
- Identify the terms:
- The highest power term: [tex]\(8x^8\)[/tex] from the first polynomial. No matching term from the second polynomial.
- Next is [tex]\(4x^7\)[/tex] from the second polynomial. No matching term from the first.
- The next powers are [tex]\(x^3\)[/tex]: Combine [tex]\(-9x^3\)[/tex] and [tex]\(6x^3\)[/tex].
- There is [tex]\(3x^2\)[/tex] from the first polynomial. No matching term from the second for [tex]\(x^2\)[/tex].
- Next is [tex]\(-2x\)[/tex] from the second polynomial. No matching term from the first for [tex]\(x\)[/tex].
- Finally, the constant term is [tex]\(+9\)[/tex] from the first polynomial. No matching constant term in the second.
3. Combine Like Terms:
- Powers of [tex]\(x^8\)[/tex]: [tex]\(8x^8\)[/tex]
- Powers of [tex]\(x^7\)[/tex]: [tex]\(4x^7\)[/tex]
- Powers of [tex]\(x^3\)[/tex]: Combine [tex]\(-9x^3\)[/tex] and [tex]\(6x^3\)[/tex] to get [tex]\(-3x^3\)[/tex].
- Powers of [tex]\(x^2\)[/tex]: [tex]\(3x^2\)[/tex]
- Powers of [tex]\(x\)[/tex]: [tex]\(-2x\)[/tex]
- Constant term: [tex]\(+9\)[/tex]
4. Write the Result as a Simplified Polynomial:
Combining all these, the result is:
[tex]\[ 8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9 \][/tex]
Therefore, the answer is:
D. [tex]\(8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9\)[/tex]
1. Write the Polynomials Clearly:
- 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. Align Like Terms:
- Look for terms that have the same power of [tex]\(x\)[/tex]. These can be combined.
- Identify the terms:
- The highest power term: [tex]\(8x^8\)[/tex] from the first polynomial. No matching term from the second polynomial.
- Next is [tex]\(4x^7\)[/tex] from the second polynomial. No matching term from the first.
- The next powers are [tex]\(x^3\)[/tex]: Combine [tex]\(-9x^3\)[/tex] and [tex]\(6x^3\)[/tex].
- There is [tex]\(3x^2\)[/tex] from the first polynomial. No matching term from the second for [tex]\(x^2\)[/tex].
- Next is [tex]\(-2x\)[/tex] from the second polynomial. No matching term from the first for [tex]\(x\)[/tex].
- Finally, the constant term is [tex]\(+9\)[/tex] from the first polynomial. No matching constant term in the second.
3. Combine Like Terms:
- Powers of [tex]\(x^8\)[/tex]: [tex]\(8x^8\)[/tex]
- Powers of [tex]\(x^7\)[/tex]: [tex]\(4x^7\)[/tex]
- Powers of [tex]\(x^3\)[/tex]: Combine [tex]\(-9x^3\)[/tex] and [tex]\(6x^3\)[/tex] to get [tex]\(-3x^3\)[/tex].
- Powers of [tex]\(x^2\)[/tex]: [tex]\(3x^2\)[/tex]
- Powers of [tex]\(x\)[/tex]: [tex]\(-2x\)[/tex]
- Constant term: [tex]\(+9\)[/tex]
4. Write the Result as a Simplified Polynomial:
Combining all these, the result is:
[tex]\[ 8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9 \][/tex]
Therefore, the answer is:
D. [tex]\(8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9\)[/tex]
