Roots of an equation of order $4$.

Question

Let $At^4+Bt^3+Ct^2+Dt+E$ and $A,B,C,D,E \in \mathbb{R}$. What can you say about the coefficients $A,B,C,D,E$ if this equation is always required to be negative (for all $t$)? What is the relation between these coefficients?

Answer 1

it factors with real numbers $p,q,r,s$ as $$ A (t^2 + pt+q)(t^2 + rt + s) $$ with $$ 4q > p^2 \; , $$ $$ 4s > r^2 \; , $$ so both $q,s > 0.$

Indeed, as they are positive reals, we can define $$ u^2 = q - \frac{p^2}{4} \; , $$ $$ v^2 = s - \frac{r^2}{4} \; , $$ and your polynomial as the negative constant $A$ times a sum of squares, $$ A \left( \left( t + \frac{p}{2} \right)^2 \left( t + \frac{r}{2} \right)^2 + v^2 \left( t + \frac{p}{2} \right)^2 + u^2 \left( t + \frac{r}{2} \right)^2 + u^2 v^2 \right) $$