Answer :
Sure, let's solve the problem step-by-step by combining the terms from the two polynomials.
### Step-by-Step Solution:
1. Write down the polynomials:
[tex]\[
\left(8x^8 - 9x^3 + 3x^2 + 9\right) + \left(4x^7 + 6x^3 - 2x\right)
\][/tex]
2. Identify and combine like terms:
- [tex]\(8x^8\)[/tex] is only present in the first polynomial.
- [tex]\(4x^7\)[/tex] is only present in the second polynomial.
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3\)[/tex] and [tex]\(+6x^3\)[/tex]:
[tex]\[
-9x^3 + 6x^3 = -3x^3
\][/tex]
- [tex]\(3x^2\)[/tex] is only present in the first polynomial.
- [tex]\(-2x\)[/tex] is only present in the second polynomial.
- The constant term, [tex]\(9\)[/tex], is only present in the first polynomial.
3. Write the polynomial with combined terms:
[tex]\[
8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9
\][/tex]
4. Compare with the given options:
The simplified polynomial matches option D:
[tex]\[
8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9
\][/tex]
Therefore, the correct answer is:
[tex]\[ D. \quad 8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9 \][/tex]
### Step-by-Step Solution:
1. Write down the polynomials:
[tex]\[
\left(8x^8 - 9x^3 + 3x^2 + 9\right) + \left(4x^7 + 6x^3 - 2x\right)
\][/tex]
2. Identify and combine like terms:
- [tex]\(8x^8\)[/tex] is only present in the first polynomial.
- [tex]\(4x^7\)[/tex] is only present in the second polynomial.
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3\)[/tex] and [tex]\(+6x^3\)[/tex]:
[tex]\[
-9x^3 + 6x^3 = -3x^3
\][/tex]
- [tex]\(3x^2\)[/tex] is only present in the first polynomial.
- [tex]\(-2x\)[/tex] is only present in the second polynomial.
- The constant term, [tex]\(9\)[/tex], is only present in the first polynomial.
3. Write the polynomial with combined terms:
[tex]\[
8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9
\][/tex]
4. Compare with the given options:
The simplified polynomial matches option D:
[tex]\[
8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9
\][/tex]
Therefore, the correct answer is:
[tex]\[ D. \quad 8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9 \][/tex]
