<s>[INST] relation between roots and coefficient in a cubic polynomial If $\alpha,\beta,\gamma$ are roots of the cubic equation
$$2x^{3}+3x^2-x-1=0$$ then I want to find the equation whose roots are $\frac{\alpha}{\beta+\gamma}, \frac{\beta}{\gamma+\alpha}, \frac{\gamma}{\alpha+\beta}$.
I have $\alpha+\beta+\gamma$= - $\frac{3}{2}$.
Therefore roots become $\frac{\alpha}{-\frac{3}{2} -\alpha}$,$\frac{\beta}{-\frac{3}{2} -\beta}$, $\frac{\gamma}{-\frac{3}{2} -\gamma}$.
To find the equation we have to find their sums and products, but it looks like complicated.
Is their any easy trick ?
 [/INST] While this is not a general method for zeroes of a cubic polynomial, this particular polynomial is amenable to familiar methods.  We can apply the Rational Zeroes Theorem to find that one zero is  $ \ \alpha \ = \ -\frac12 \ \ , $ so we may factor $ \ 2x^3 + 3x^2 - x - 1 \ = \ \left(x + \frac12 \right)·2·(x^2 + x - 1) \ = \ 0 \ \ . $  This polynomial also has the helpful property that it has symmetry about $ \ x \ = \ -\frac12 \ \ : \ 2·\left(u - \frac12 \right)^3 + 3·\left(u - \frac12 \right)^2 - \left(u - \frac12 \right) - 1 $ $ = \ 2u·\left(u^2 - \frac54 \right) \ \ . $  The other zeroes of the original polynomial are then $ \ \beta \ , \ \gamma \ = \ \alpha \ \pm \ \Delta \ = \ -\frac12  \pm  \frac{\sqrt5}{2} \ \ . \ $  (We will have need of $ \ \Delta^2 \ = \ \frac54 \ $ shortly. These two zeroes are in fact $ \ \frac{1}{\phi} \ $ and $ \ -\phi \ \ , $ with $ \ \phi \ $ being the "Golden Ratio", but we won't be making use of that in this discussion.)
We seek a cubic polynomial with the transformed zeroes $$ \alpha' \  = \ \frac{\alpha}{\beta \ + \ \gamma} \  = \ \frac{-1/2}{-1} \ = \ \frac12 \ \ \ , \ \ \ \beta' \ = \ \frac{\beta}{\alpha \ + \ \gamma} \  = \ \frac{\alpha \ + \ \Delta}{2\alpha \ - \ \Delta} \ = \ \frac{1 \ + \ 2\Delta}{2 \ - \ 2\Delta} \ \ ,  $$
$$\gamma' \ = \ \frac{\gamma}{\alpha \ + \ \beta} \  = \ \frac{\alpha \ - \ \Delta}{2\alpha \ + \ \Delta} \ = \ \frac{1 \ - \ 2\Delta}{2 \ + \ 2\Delta} \ \ .  $$
From these, we compute the coefficients of a new monic polynomial from
$$ c' \ \ = \ \ -(\alpha'  ·  \beta'  ·  \gamma') \ \ = \ \ -\frac{(1 \ + \ 2\Delta)·(1 \ - \ 2\Delta)}{2 ·(2 \ - \ 2\Delta)·(2 \ + \ 2\Delta)} \ \ = \ \ \frac{4\Delta^2 \ - \ 1}{8 · (1 \ - \ \Delta^2)} $$ $$ = \ \ \frac{5 \ - \ 1}{-2} \ \ = \ \ -2 \ \ ; $$
$$ a' \ \ = \ \ -(\alpha' \ + \ \beta' \ + \ \gamma') \ \ = \ \ - \left[ \ \frac12 \ + \  \frac{1 \ + \ 2\Delta}{2 \ - \ 2\Delta} \ + \  \frac{1 \ - \ 2\Delta}{2 \ + \ 2\Delta} \ \right] \ \ = \ \ \frac32· \left(\frac{\Delta^2 \ + \ 1}{  \Delta^2 \ - \ 1} \right) $$ $$ = \ \ \frac32· \left(\frac{5 \ + \ 4}{  5 \ - \ 4} \right) \ \ = \ \ \frac{27}{2} \ \ ; $$
$$ \frac{b'}{c'} \ \ = \  \ \frac{\alpha' · \beta' \ +  \ \alpha' · \gamma' \ + \ \beta' · \gamma'}{- \ (\alpha'  · \beta'  ·  \gamma')} \ \ = \ \ - \left(\frac{1}{\alpha'} \ + \frac{1}{\beta'} \ + \ \frac{1}{\gamma'} \right) $$
$$ = \ \ - \left[ \ 2 \ + \  \frac{2 \ - \ 2\Delta}{1 \ + \ 2\Delta} \ + \  \frac{2 \ + \ 2\Delta}{1 \ - \ 2\Delta} \ \right] \ \ = \ \ \frac{6}{  4\Delta^2 \ - \ 1} \ \ = \ \ \frac{6}{ 5 \ - \ 1} \ \ = \ \ \frac32 $$
$$ \Rightarrow \ \ b' \ \ = \ \ \frac32 \ · \ (-2) \ \ = \ \ -3 \ \ . $$
A cubic polynomial having the transformed zeroes which has integer coefficients is then $ \ 2x^3 \ + \ 27x^2 \ - \ 6x \ - \ 4 \ \ . $
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