TITLE: roots of the polynomial equations and relation among the coefficients
QUESTION [5 upvotes]: If the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$ ($a,b,c$ are real numbers) has no real roots and if at least one root is of modulus one, then what is the relation between $a,b$ and $c$?

REPLY [3 votes]: Factoring $x^4+ax^3+bx^2+cx+1=(x^2+dx+1)(x^2+ex+1)$ shows that $a=c=d+e$ and $b=2+de$.  To require no real roots we have the inequalities $-2<d,e<2$.  Hence all possible triplets $(a,b,c)$ have the form:
$$(a,b,c)=(d+e,2+de,d+e)$$
for some $(d,e)\in(-2,2)\times (-2,2)$.
Intuitively, $d$ is twice the real part of one pair of complex roots, $z,\bar{z}$.  Since $|z|=1$, and $z\ne\bar{z}$, it follows that $-1<\operatorname{Re}z<1$, which gives us our inequalities for $d$.  Similarly for $e$.