+349 Let r, s, and t be solutions of the equation 3x^3 - 4x^2 - 2x + 12 = 0. Compute
\frac{rs}{t^2} + \frac{rt}{s^2} + \frac{st}{r^2}.
+761 Using Vieta’s formulas:
r + s + t = 4/3
rs + rt + st = −2/3
rst = −4
Then, rewriting each term using rst = −4:
rs / t2 = -4 / t^3, rt / s2 = -4 / s^3, and st / r2 = -4 / r^3
Because of this, the sum simplifies to:
−4(1 / r^3 + 1 / s^3 + 1 / t^3)
Afterwards, using the identity:
1 / r^3 + 1 / s^3 + 1 / t^3 = ((rs + rt + st)(r + s + t) − rst) / (rst)2
Finally, substituting the values:
(−23)(43) − (−4)) / 16 = 28 / 144 = 7 / 36
And then, multiplying -4:
−28 / 36 = −7 / 9
Therefore, the final answer is:
-7 / 9
+761 Using Vieta’s formulas:
r + s + t = 4/3
rs + rt + st = −2/3
rst = −4
Then, rewriting each term using rst = −4:
rs / t2 = -4 / t^3, rt / s2 = -4 / s^3, and st / r2 = -4 / r^3
Because of this, the sum simplifies to:
−4(1 / r^3 + 1 / s^3 + 1 / t^3)
Afterwards, using the identity:
1 / r^3 + 1 / s^3 + 1 / t^3 = ((rs + rt + st)(r + s + t) − rst) / (rst)2
Finally, substituting the values:
(−23)(43) − (−4)) / 16 = 28 / 144 = 7 / 36
And then, multiplying -4:
−28 / 36 = −7 / 9
Therefore, the final answer is:
-7 / 9
