[Math] Condition For 4th degree polynomial equation having positive roots

Question

Consider the biquadratic polynomial equation $\rho_0y^4+\rho_1y^3+\rho_2y^2+\rho_3y+\rho_4y=0$, where $\rho_0,\rho_1,\rho_2, \rho_4$ are positive and $\rho_3$ is negative. So by Descartes' rule of signs it has either two positive roots or no positive root.

Now what is the necessary condition (on the coefficient $\rho_i, i=0,1,..4$ ) for which the above biquadratic equation must have two positive roots??

Answer 1

Let $f(z)=ax^4+bx^3+cx^2+dx+e$ be a quartic polynomial with real coefficients and $a>0$, $e>0$.

Define:

$\alpha = ba^{-3/4}e^{-1/4}$

$\beta=ca^{-1/2}e^{-1/2}$

$\gamma=da^{-1/4}e^{-3/4}$

Then $f(x)\geq0$ for all $x>0$ the following conditions are sufficient for positivity:

• $\alpha>-\dfrac{\beta +2}{2}$ and $\gamma>-\dfrac{\beta +2}{2}$ for $\beta\leq6$
• $\alpha>-2\sqrt{\beta -2}$ and $\gamma>-2\sqrt{\beta -2}$ for $\beta>6$

Hope this help you.

Regards

Ric