+452 Let r, s, and t be solutions of the equation x^3 + 2x^2 - 5x + 15 = 0. Compute
\frac{1}{r - 2s - 2t} + \frac{1}{s - 2r - 2t} + \frac{1}{t - 2r - 2s}
+130474 \(\frac{1}{r - 2s - 2t} + \frac{1}{s - 2r - 2t} + \frac{1}{t - 2r - 2s} \)
r + s + t = -2
2r + 2s + 2t = -4
2r = -4 - 2s - 2t
2r + 4 = -2 -2t
And
2s + 4 = -2r -2t
2t + 4 = -2r - 2s
And
rs + rt + st = -5
rst = -15
Simplifying, we have
1 + 1 + 1
_________ ______ ______ =
3r + 4 3s +4 3t + 4
(3s + 4) + (3t + 4) + ( 3s + 4)
________________________ =
(3r + 4) (3s + 4) (3t + 4)
3(r + s + t) + 12
____________________________________ =
27rst + 36 (rs + rt + st) + 48 ( r + s + t) + 64
3 (-2) + 12
________________________ =
27(-15) + 36(-5) + 48(-2) + 64
- 6
_______
617
