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


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 $a$? Did you mean $f$ instead of $a$? – copper.hat Nov 23 '12 at 19:11
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. – André Nicolas Nov 23 '12 at 19:12
up vote 3 down vote accepted

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 $$\alpha+\beta + \gamma = 0$$

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