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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}.

 Feb 25, 2025

Best Answer 

 #1
avatar+761 
+1

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

 Feb 25, 2025
 #1
avatar+761 
+1
Best Answer

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

history Feb 25, 2025

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