Subtract the polynomials: (9n^3 - n^7 - 9) - (8n^3 - n^7 - 11). Simplify your answer and do not factor.

Step-by-step solution

1. 1

   Distribute the minus sign across every term of the second bracket. This is where nearly all the marks are lost — the sign flips on all three terms, including the one that is already negative:

   $-\left(8n^3-n^7-11\right)=-8n^3+n^7+11.$

2. 2

   Rewrite the whole expression with no brackets left.

   $9n^3-n^7-9-8n^3+n^7+11.$

3. 3

   Group like terms by degree. Sort by the exponent on $n$ so nothing gets stranded:

   $\underbrace{\left(-n^7+n^7\right)}_{\text{degree }7}+\underbrace{\left(9n^3-8n^3\right)}_{\text{degree }3}+\underbrace{\left(-9+11\right)}_{\text{constants}}.$

4. 4

   Combine each group. The degree-7 terms are exact opposites and vanish, which is why the answer is only cubic:

   $0+n^3+2=n^3+2.$

5. 5

   Check with a test value. At $n=2$ the original is $(72-128-9)-(64-128-11)=-65-(-75)=10$, and $n^3+2=8+2=10$. At $n=-1$ the original is $(-9+1-9)-(-8+1-11)=-17-(-18)=1$, and $(-1)^3+2=1$. Both agree, so the difference is $n^3+2$.