Answer :
To solve the given problem, we need to add the two polynomials together. Here's a step-by-step guide:
1. Identify 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. Adding the Polynomials:
- We add the corresponding terms from each polynomial. Remember that only like terms (terms with the same power of [tex]\(x\)[/tex]) should be added together.
3. List the Terms by Degree:
- From the first polynomial, we have:
- [tex]\( 8x^8 \)[/tex]
- [tex]\( -9x^3 \)[/tex]
- [tex]\( 3x^2 \)[/tex]
- [tex]\( +9 \)[/tex] (constant term)
- From the second polynomial, we have:
- [tex]\( 4x^7 \)[/tex]
- [tex]\( 6x^3 \)[/tex]
- [tex]\( -2x \)[/tex]
4. Combine Like Terms:
- Powers of [tex]\(x\)[/tex] with degree 8: [tex]\( 8x^8 \)[/tex] is available with no other terms to combine. So, we have [tex]\( 8x^8 \)[/tex].
- Powers of [tex]\(x\)[/tex] with degree 7: [tex]\( 4x^7 \)[/tex] is available with no other terms to combine. So, we have [tex]\( 4x^7 \)[/tex].
- Powers of [tex]\(x\)[/tex] with degree 3: Combine [tex]\( -9x^3 \)[/tex] and [tex]\( +6x^3 \)[/tex]. This results in [tex]\( -9x^3 + 6x^3 = -3x^3 \)[/tex].
- Powers of [tex]\(x\)[/tex] with degree 2: There is only [tex]\( 3x^2 \)[/tex] from the first polynomial.
- Powers of [tex]\(x\)[/tex] with degree 1: There is only [tex]\( -2x \)[/tex] from the second polynomial.
- Constant terms: The constant from the first polynomial is [tex]\( +9 \)[/tex].
5. Write the Resulting Polynomial:
- Combine all the terms you've calculated: [tex]\( 8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9 \)[/tex].
The final polynomial is therefore:
[tex]\[ 8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9 \][/tex]
This matches option D:
D. [tex]\( 8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9 \)[/tex]
1. Identify 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. Adding the Polynomials:
- We add the corresponding terms from each polynomial. Remember that only like terms (terms with the same power of [tex]\(x\)[/tex]) should be added together.
3. List the Terms by Degree:
- From the first polynomial, we have:
- [tex]\( 8x^8 \)[/tex]
- [tex]\( -9x^3 \)[/tex]
- [tex]\( 3x^2 \)[/tex]
- [tex]\( +9 \)[/tex] (constant term)
- From the second polynomial, we have:
- [tex]\( 4x^7 \)[/tex]
- [tex]\( 6x^3 \)[/tex]
- [tex]\( -2x \)[/tex]
4. Combine Like Terms:
- Powers of [tex]\(x\)[/tex] with degree 8: [tex]\( 8x^8 \)[/tex] is available with no other terms to combine. So, we have [tex]\( 8x^8 \)[/tex].
- Powers of [tex]\(x\)[/tex] with degree 7: [tex]\( 4x^7 \)[/tex] is available with no other terms to combine. So, we have [tex]\( 4x^7 \)[/tex].
- Powers of [tex]\(x\)[/tex] with degree 3: Combine [tex]\( -9x^3 \)[/tex] and [tex]\( +6x^3 \)[/tex]. This results in [tex]\( -9x^3 + 6x^3 = -3x^3 \)[/tex].
- Powers of [tex]\(x\)[/tex] with degree 2: There is only [tex]\( 3x^2 \)[/tex] from the first polynomial.
- Powers of [tex]\(x\)[/tex] with degree 1: There is only [tex]\( -2x \)[/tex] from the second polynomial.
- Constant terms: The constant from the first polynomial is [tex]\( +9 \)[/tex].
5. Write the Resulting Polynomial:
- Combine all the terms you've calculated: [tex]\( 8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9 \)[/tex].
The final polynomial is therefore:
[tex]\[ 8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9 \][/tex]
This matches option D:
D. [tex]\( 8x^8 + 4x^7 - 3x^3 + 3x^2 - 2x + 9 \)[/tex]
