Roots of cubic polynomial lying inside the circle Show that all roots of $a+bz+cz^2+z^3=0$ lie inside the circle $|z|=max{\\{1,|a|+|b|+|c| \\}}$ Now this problem is given in Beardon's Algebra and Geometry third chapter on complex numbers. What might be relevant for this problem: * author previously discussed roots of unity; * a little (I mean abut a page of informal discussion) about cubic and quartic equations; * then gave proof of fundamental theorem of algebra (the existence of root was given as a informal proof and rest using induction) and then the corollary of it (If $p(z) = q(z)$ at $n + 1$ distinct points then $p(z) = q(z)$ for all $z$, where both polynomials are of degree at most $n$); I was trying to see how I should approach it with no success for quite some time. Looked on the net and found, that this would be kind of easy with Rouche's theorem, but I was not given that. So is it possible to solve it in a simple way with what was given? Thanks!

Suppose $a + bz + cz^2 + z^3 = 0$ and $|z| > \max\\{1,|a|+|b|+|c|\\}$. Use the triangle inequality to show that

$$ |z^3| = |a + bz + cz^2| < (|a|+|b|+|c|)|z^2|, $$

which contradicts the assumption on the size of $|z|$.