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