good transformation equation question (1 Viewer)

[barbernator]

here's a good question from my test today if you want to have a crack at it, its not very hard if you find the trick.

<a href="http://www.codecogs.com/eqnedit.php?latex=the~equation~x^3-kx^2@plus;k^2x-1=0~has~roots~a,b,c\\ find~and~equation~with~the~roots~\frac{a}{bc},\frac{b}{ac},\frac{c}{ab}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?the~equation~x^3-kx^2+k^2x-1=0~has~roots~a,b,c\\ find~and~equation~with~the~roots~\frac{a}{bc},\frac{b}{ac},\frac{c}{ab}" title="the~equation~x^3-kx^2+k^2x-1=0~has~roots~a,b,c\\ find~and~equation~with~the~roots~\frac{a}{bc},\frac{b}{ac},\frac{c}{ab}" /></a>
for some reason for me, the middle root is stuffing up, it should be b/ac

 

[jet]

Let's see if I can remember how to do these...



For the people who don't know: WHY did I choose that transformation?

Each new root is of the form


And my imperative is to get it into a form that I can express as a ratio of coefficients in the polynomial. The only relationships between roots that I have are the sum of roots, the pairwise sum of roots and the product of roots. So, the simplest form is:



One of the roots cancels from the numerator and denominator to give you the original form of the root.

So my transform must be



and the product of roots is 1 from the equation, giving me my transformation.

 

[lolcakes52]

Okay, I hope I dont sound stupid but I dont understand how cabc takes the form of root/product of two other roots. Am i being incredibly dumb?

edit: yes i was, misread the question. Im such a dumbass.