Please explain the formula for the sum of the cubes and the difference: $a^3 - b^3$ and $a^3 + b^3$? I have not yet fully mastered the formula for accelerated multiplication. I was able to understand everything except the last two. I would like to deal with them. Why have : $$ a^3 + b^3 = (a+b)(a^2 -ab + b^2)^* $$ and $$ a^3 - b^3 = (a-b)(a^2 + ab + b^2) $$ Can someone paint me a formula in detail. Why is this the way it works.

Let $x=a/b$.  We want to factor $b^3(x^3\pm1)$. Note that $1$ is a root of $x^3-1$, so $x-1$ is a factor of $x^3-1$.  You could find the quotient by polynomial long division.  Also, $-1$ is a root of $x^3+1$, so $x+1$ is a factor of $x^3+1$, and again you could find the quotient by polynomial long division.  

(From this, it is easy to see why $a^n+b^n$ has that kind of factorization only for $n$ odd.)
