Factoring within a proof In the proof text I am using, I am trying to understand a proof of the fact that the geometric mean is less than or equal to the arithmetic mean by showing that:
rst $\le$ (r$^3$ + s$^3$ + t$^3$)/3
The answer in the back says to note that:
r$^3$ + s$^3$ + t$^3$ - 3rst = $\frac 12$(r + s + t)[(r - s)$^2$ + (s - t)$^2$ + (t - r)$^2$]
That said, I have no idea how they got the right side from the left, let alone how to continue with the proof. Does anyone have any pointers as to how to begin factoring the left to get the right?
Thanks!
Chris

$$r^3+s^3+t^3-3trs=r^3+3r^2s+3rs^2+s^3+t^3-3r^2s-3rs^2-3rts=$$
$$=(r+s)^3+t^3-3rs(r+s+t)=(r+s+t)((r+s)^2-(r+s)t+t^2)-3rs(r+s+t)=$$
$$(r+s+t)(r^2+2rs+s^2-rt-st-3ts)=$$
$$=(r+s+t)(r^2+s^2+t^2-rs-rt-st)=\frac{1}{2}(r+s+t)((r-s)^2+(r-t)^2+(s-t)^2).$$
