Question

Show that the distance of the roots of the equation $\left|sin{\theta }_1\right|z^3+\left|sin{\theta }_2\right|z^2+\left|sin{\theta }_3\right|z+\left|sin{\theta }_4\right|=3$ from $z=0$ is greater than 2/3.

Answer 1

We know that $\left|sin{\theta }_k\right|<1$. Given,
$\left|sin{\theta }_1\right|z^3+\left|sin{\theta }_2\right|z^2+\left|sin{\theta }_3\right|z+\left|sin{\theta }_4\right|=3$
or $\left|3\right|=|\left|sin{\theta }_1\right|z^3+\left|sin{\theta }_2\right|z^2+\left|sin{\theta }_3\right|z+\left|sin{\theta }_4\right||\le 1|z^3+z^2+z+1|$

$<{\left|z\right|}^3+{\left|z\right|}^2+\left|z\right|+1<1+\left|z\right|+{\left|z\right|}^2+{\left|z\right|}^3+{\left|z\right|}^4+...\infty$ ($\therefore \left|z\right|<1$)
or $3<\frac{1}{1-\left|z\right|}$
or $3-3\left|z\right|<1$
or $2<3\left|z\right|$
or $\left|z\right|>\frac{2}{3}$

Ans: 1