5. Multiply

(*a* + *b*)(*a*^{2} - *ab* + *b*^{2})
Multiply each term in the first polynomial by each term in the other polynomial.

and the answer is

*a*^{3} + *b*^{3}
This illustrates that a sum of odd powers always factors. One of the factors of *a*^{n} + *b*^{n} is *a* + *b*. The first term in the other factor is *a*^{n-1} and each subsequent term has one less factor of *a* and one more factor of *b*, until you get to *b*^{n-1}. When factoring a sum of odd powers, the signs on each of these terms alternates. That is the reason it only works with sums of odd powers. With sums of odd powers, the last term is + *b*^{n-1}.