If we put a7+b7=0, then we know a=-b is a solution, that is, a+b is a factor. If we divide by this factor we will end up with another factor. The easiest way to divide is to use synthetic division with -b as the divisor:
-b | 1 0 0 0 0 0 0 b7
1 -b b2 -b3 b4 -b5 b6 -b7
1 -b b2 -b3 b4 -b5 b6 | 0
which translates to the polynomial a6-a5b+a4b2-a3b3+a2b4-ab5+b6.
a7+b7=(a+b)(a6-a5b+a4b2-a3b3+a2b4-ab5+b6).