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

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Answer 1 · 2015-04-08T20:35:48

#z^4 + z + 2= 0#

#z^4 + z = -2#

#abs(z^4 + z) =abs(- 2)=2#

#abs(z^4 + z) = absz abs(z^3+1)#

If #absz <1#, then #absz^3 <1#,

And #abs(z^3+1) <= abs(z^3)+abs1 <1+1=2#

Finally If #absz <1#, then

#abs(z^4 + z) =absz abs(z^3+1)<1*2<2# so we cannot have

#z^4 + z = -2#

#abs(z^4 + z) =abs(- 2)=2# as required for a solution.

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