To solve the polynomial subtraction [tex]\((5x^6 + 9x^8 + 6 + 9x^7) - (8 - 2x^7 + 6x^8 - 3x^6)\)[/tex], we need to perform the subtraction term by term. Here is a step-by-step solution:
1. Rewrite the polynomials with their coefficients aligned:
- [tex]\(A(x) = 5x^6 + 9x^8 + 6 + 9x^7\)[/tex]
- [tex]\(B(x) = 8 - 2x^7 + 6x^8 - 3x^6\)[/tex]
2. Distribute the negative sign for the second polynomial:
- [tex]\(A(x) - B(x) = (5x^6 + 9x^8 + 6 + 9x^7) - (8 - 2x^7 + 6x^8 - 3x^6)\)[/tex]
- [tex]\(= 5x^6 + 9x^8 + 6 + 9x^7 - 8 + 2x^7 - 6x^8 + 3x^6\)[/tex]
3. Combine like terms:
- For [tex]\(x^8\)[/tex]: [tex]\(9x^8 - 6x^8 = 3x^8\)[/tex]
- For [tex]\(x^7\)[/tex]: [tex]\(9x^7 + 2x^7 = 11x^7\)[/tex]
- For [tex]\(x^6\)[/tex]: [tex]\(5x^6 + 3x^6 = 8x^6\)[/tex]
- Constant term: [tex]\(6 - 8 = -2\)[/tex]
4. Write the simplified polynomial:
- [tex]\(3x^8 + 11x^7 + 8x^6 - 2\)[/tex]
Thus, the correct answer is:
C. [tex]\(3x^8 + 11x^7 + 8x^6 - 2\)[/tex]
