I'm trying to prove that the cubic equation
$_3 ^3 + _2 ^2 + _1 + _0=0$
has three real roots. The coefficients are
$_3=−1−−−$
$_2=−2(++)$
$_1=(+)+(−3)$
$_0=2$
where each of , and are greater than zero. Applying the Rule of Descartes indicates that the polynomial has one real positive root and zero or two real negative roots. I've played around with forming the discriminant for the cubic, and a 3-D contour plot in Mathematica suggests that the discriminant is positive for ,,>0 (indicating three real roots). However, I've hit a wall trying to come up with a formal proof that applies for any (positive) choice of ,,.
One thing of note is that the polynomial coefficients are unchanged on interchange of and , and so the discriminant is a symmetric function of and .