# z  is a complex number. Show that the equation z^4 + z + 2= 0 cannot have a root z such that |z|<1?

Apr 8, 2015

${z}^{4} + z + 2 = 0$

${z}^{4} + z = - 2$

$\left\mid {z}^{4} + z \right\mid = \left\mid - 2 \right\mid = 2$

$\left\mid {z}^{4} + z \right\mid = \left\mid z \right\mid \left\mid {z}^{3} + 1 \right\mid$

If $\left\mid z \right\mid < 1$, then ${\left\mid z \right\mid}^{3} < 1$,

And $\left\mid {z}^{3} + 1 \right\mid \le \left\mid {z}^{3} \right\mid + \left\mid 1 \right\mid < 1 + 1 = 2$

Finally If $\left\mid z \right\mid < 1$, then

$\left\mid {z}^{4} + z \right\mid = \left\mid z \right\mid \left\mid {z}^{3} + 1 \right\mid < 1 \cdot 2 < 2$ so we cannot have

${z}^{4} + z = - 2$

$\left\mid {z}^{4} + z \right\mid = \left\mid - 2 \right\mid = 2$ as required for a solution.

(There may be more elegant proofs, but this works.)