which polynomial correctly combines the like terms and expresses the given polynomial in standard form…

which polynomial correctly combines the like terms and expresses the given polynomial in standard form? 8mn^5 - 2m^6 + 5m^2n^4 - m^3n^3 + n^6 - 4m^6 + 9m^2n^4 - mn^5 - 4m^3n^3\nn^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3 - 6m^6\n-2m^6 - 5m^3n^3 + 14m^2n^4 + 7mn^5 + n^6\n14m^2n^4 + 7mn^5 - 6m^6 - 5m^3n^3 + n^6\nn^6 - 6m^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3

which polynomial correctly combines the like terms and expresses the given polynomial in standard form? 8mn^5 - 2m^6 + 5m^2n^4 - m^3n^3 + n^6 - 4m^6 + 9m^2n^4 - mn^5 - 4m^3n^3\nn^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3 - 6m^6\n-2m^6 - 5m^3n^3 + 14m^2n^4 + 7mn^5 + n^6\n14m^2n^4 + 7mn^5 - 6m^6 - 5m^3n^3 + n^6\nn^6 - 6m^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3

Answer

Explanation:

Step1: Combine like - terms of $mn^{5}$

$8mn^{5}-mn^{5}=(8 - 1)mn^{5}=7mn^{5}$

Step2: Combine like - terms of $m^{6}$

$-2m^{6}-4m^{6}=(-2-4)m^{6}=-6m^{6}$

Step3: Combine like - terms of $m^{2}n^{4}$

$5m^{2}n^{4}+9m^{2}n^{4}=(5 + 9)m^{2}n^{4}=14m^{2}n^{4}$

Step4: Combine like - terms of $m^{3}n^{3}$

$-m^{3}n^{3}-4m^{3}n^{3}=(-1-4)m^{3}n^{3}=-5m^{3}n^{3}$

Step5: Write the polynomial in standard form

The standard form of a polynomial arranges the terms in descending order of the sum of the exponents of the variables. The sum of exponents for each term:

  • For $n^{6}$, sum of exponents is 6.
  • For $-6m^{6}$, sum of exponents is 6.
  • For $7mn^{5}$, sum of exponents is $1 + 5=6$.
  • For $14m^{2}n^{4}$, sum of exponents is $2+4 = 6$.
  • For $-5m^{3}n^{3}$, sum of exponents is $3 + 3=6$. We usually order alphabetically when the sum of exponents is the same. So the polynomial is $n^{6}-6m^{6}+7mn^{5}+14m^{2}n^{4}-5m^{3}n^{3}$

Answer:

$n^{6}-6m^{6}+7mn^{5}+14m^{2}n^{4}-5m^{3}n^{3}$