[Math] roots of a cubic polynomial

Question

Consider a cubic polynomial of the form

$$f(x)=a_3x^3+a_2x^2+a_1x+a_0$$

where the coefficients are non-zero reals. Conditions for which this equation has three real simple roots are well-known. What conditions would guarantee that none of these roots is positive? In other words, what constraints on the parameters would guarantee that the polynomial has no positive roots? Please provide references also, if possible.

Answer 1

Using Routh–Hurwitz:

• if $a_0, a_1, a_2, a_3 > 0$, you arrive at the condition $a_2 a_1 > a_3 a_0$.
• if $a_0, a_1, a_2, a_3 < 0$, you also arrive at the condition $a_2 a_1 > a_3 a_0$.
• if $a_3$ and $a_2$ have different signs, then the polynomial has positive roots.