Find the value of rs/t+st/r+tr/s where r,s, and t are the roots of the equation x^3-5x^2+6x-9=0. I think it has something to do with Vieta's formulas, but I can't easily manipulate the product or some to find it. Can someone help?
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When r,s,t, are roots of the an eqn,(x-r)(x-s)(x-t) = 0
x^3 -(r + s + t)x^2 +(rs + rt + st)x -rst = 0
Comparing,co-efficients we get:
r + s + t = 5
rs + rt +st = 6
rst = 9
Now,rs/t + rt/s + st/r = ((rs)^2 + (rt)^2 + (st)^2)/rst
= ((rs + rt +st)^2 - 2(r^2st + s^2rt + t^2st))/rst
= 36/9 - 2rst(r + s + t)/rst
= 4 - 2 * 5
= -6
x^3 -(r + s + t)x^2 +(rs + rt + st)x -rst = 0
Comparing,co-efficients we get:
r + s + t = 5
rs + rt +st = 6
rst = 9
Now,rs/t + rt/s + st/r = ((rs)^2 + (rt)^2 + (st)^2)/rst
= ((rs + rt +st)^2 - 2(r^2st + s^2rt + t^2st))/rst
= 36/9 - 2rst(r + s + t)/rst
= 4 - 2 * 5
= -6