To determine which polynomial correctly combines the like terms and expresses the given polynomial in standard form, let's go through each step carefully.
We're starting with the 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]
First, let's combine like terms:
1. Combine the terms with [tex]\(mn^5\)[/tex]:
[tex]\[ 8mn^5 - mn^5 = 7mn^5 \][/tex]
2. Combine the terms with [tex]\(m^2n^4\)[/tex]:
[tex]\[ 5m^2n^4 + 9m^2n^4 = 14m^2n^4 \][/tex]
3. Combine the terms with [tex]\(m^6\)[/tex]:
[tex]\[ -2m^6 - 4m^6 = -6m^6 \][/tex]
4. Combine the terms with [tex]\(m^3n^3\)[/tex]:
[tex]\[ -m^3n^3 - 4m^3n^3 = -5m^3n^3 \][/tex]
5. The term [tex]\(n^6\)[/tex] appears by itself, so it stays as [tex]\(n^6\)[/tex].
Now, we can write all the combined terms together in standard form (arranging the terms in order of the powers of the variables):
[tex]\[ n^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3 - 6m^6 \][/tex]
So, the polynomial that correctly combines the like terms and expresses it in standard form is:
[tex]\[ n^6 - 6m^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3 \][/tex]
Therefore, the correct option is:
[tex]\[ n^6 - 6m^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3 \][/tex]
