Question

Which polynomial correctly combines the like terms and expresses the given polynomial in standard form? 8mn5 – 2m6 + 5m2n4 – m3n3 + n6 – 4m6 + 9m2n4 – mn5 – 4m3n3 n6 + 7mn5 + 14m2n4 – 5m3n3 – 6m6 –2m6 – 5m3n3 + 14m2n4 + 7mn5 + n6 14m2n4 + 7mn5 – 6m6 – 5m3n3 + n6 n6 – 6m6 + 7mn5 + 14m2n4 – 5m3n3

Answer 1

The correct combination and order would be:  $-6m^6-5m^3n^3+14m^2n^4+7mn^5+n^6$

Our original polynomial looks like this:
$8mn^5-2m^6+5m^2n^4-m^3n^3+n^6-4m^6+9m^2n^4-mn^5-4m^3n^3$

We will start with the first term.  We want to find any other terms that have $mn^5$, since like terms have the same variables with the same exponents:
$\bold{8mn^5}-2m^6+5m^2n^4-m^3n^3+n^6-4m^6+9m^2n^4\bold{-mn^5}-4m^3n^3$

Combining the two bold terms, we have 
$7mn^5-2m^6+5m^2n^4-m^3n^3+n^6-4m^6+9m^2n^4-4m^3n^3$

Focusing on the second term now:
$7mn^5\bold{-2m^6}+5m^2n^4-m^3n^3+n^6\bold{-4m^6}+9m^2n^4-4m^3n^3$

Combining the bold terms gives us:
$7mn^5-6m^6+5m^2n^4-m^3n^3+n^6+9m^2n^4-4m^3n^3$

Focus on the third term:
$7mn^5-6m^6+\bold{5m^2n^4}-m^3n^3+n^6+\bold{9m^2n^4}-4m^3n^3$

Combining the bold terms:
$7mn^5-6m^6+14m^2n^4-m^3n^3+n^6-4m^3n^3$

Focus on the fourth term:
$7mn^5-6m^6+14m^2n^4\bold{-m^3n^3}+n^6\bold{-4m^3n^3}$

Combining bold terms:
$7mn^5-6m^6+14m^2n^4-5m^3n^3+n^6$

Our last step is to order them by variable and by exponent.  Start with the largest exponent of m down to the smallest, then up to the largest exponent of n:
$-6m^6-5m^3n^3+14m^2n^4+7mn^5+n^6$