Newton's Identity, Lesson 4.2: An AIME Problem, cubic sums

2008 AIME II, Problem 7

Let $r$, $s$, and $t$ be the three roots of the equation\[8x^3 + 1001x + 2008 = 0.\]Find $(r + s)^3 + (s + t)^3 + (t + r)^3$.