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I've a problem in factorization of this parametric polynomial.

$t^3-2bfc-t(c^2+f^2+b^2)=0$

To be onest I'm just interested in proving that $\exists! \space (f,b,c)$ such that roots are all real and positive.

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    What is $a$? I take it you by $\exists!$ you mean there exist unique $a$, $b$, $c$ such that $\dots$. That seems unlikely, given the continuous dependence of roots on the parameters.2012-11-23

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I assume your $f$ is same as $a$. If the roots of $t^3 - (a^2 + b^2 + c^2)t - 2abc=0$ are $\alpha, \beta$ and $\gamma$, then the sum of the roots is the coefficient of $t^2$, which in this case is $0$. Hence, $\alpha + \beta + \gamma = 0$ Hence, if all roots are real and positive, then $\alpha + \beta + \gamma > 0$ contradicting the fact that $\alpha+\beta + \gamma = 0$