To determine which polynomial correctly combines the like terms and expresses the given polynomial in standard form, let's break down the problem step-by-step.
1. Start with the given polynomial:
[tex]\[
8mn^5 - 2m^6 + 5m^2n^4 - m^3n^3 + n^6 - 4m^6 + 9m^2n^4 - mn^5 - 4m^3n^3
\][/tex]
2. Identify and group the like terms:
- [tex]\(mn^5\)[/tex]: [tex]\(8mn^5\)[/tex] and [tex]\(-mn^5\)[/tex]
- [tex]\(m^6\)[/tex]: [tex]\(-2m^6\)[/tex] and [tex]\(-4m^6\)[/tex]
- [tex]\(m^2n^4\)[/tex]: [tex]\(5m^2n^4\)[/tex] and [tex]\(9m^2n^4\)[/tex]
- [tex]\(m^3n^3\)[/tex]: [tex]\(-m^3n^3\)[/tex] and [tex]\(-4m^3n^3\)[/tex]
- [tex]\(n^6\)[/tex]: [tex]\(n^6\)[/tex]
3. Combine the coefficients of the like terms:
- For [tex]\(mn^5\)[/tex]: [tex]\(8 - 1 = 7\)[/tex]
- For [tex]\(m^6\)[/tex]: [tex]\(-2 - 4 = -6\)[/tex]
- For [tex]\(m^2n^4\)[/tex]: [tex]\(5 + 9 = 14\)[/tex]
- For [tex]\(m^3n^3\)[/tex]: [tex]\(-1 - 4 = -5\)[/tex]
- For [tex]\(n^6\)[/tex]: [tex]\(1\)[/tex]
4. Construct the simplified polynomial:
[tex]\[
n^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3 - 6m^6
\][/tex]
5. Match this polynomial with the given options:
[tex]\[
n^6 - 6m^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3
\][/tex]
So, the polynomial that correctly combines the like terms and expresses the given polynomial in standard form is:
[tex]\[
n^6 - 6m^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3
\][/tex]
Hence, the correct answer is:
[tex]\[
n^6 - 6m^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3
\][/tex]
