How can you factor a^7 + b^7 completely?

Question

factor completely. Pratice of factoring. Sum of odd powers.

Answer 1

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).