# Math Help - Complex numbers problem

1. ## Complex numbers problem

Show that z^4 - 2z3 + 6z^2 -8z + 8 =0 has a root of the form ki, where k is real. Hence solve the equation z^4 - 2z3 + 6z^2 -8z + 8 =0

my steps are
let f(x) = z^4 - 2z3 + 6z^2 -8z + 8
since, ki is a root
f(ki)= z^4 - 2z3 + 6z^2 -8z + 8 = 0
substituting ki inside,
i get (k^2 -2)(k^2 -4) +2k(k^2 -4)i =0

im not sure how to proceed at this point..

the answer is z=2i, -2i, 1+i or 1-i

2. Originally Posted by mephisto50
Show that z^4 - 2z3 + 6z^2 -8z + 8 =0 has a root of the form ki, where k is real. Hence solve the equation z^4 - 2z3 + 6z^2 -8z + 8 =0

my steps are
let f(x) = z^4 - 2z3 + 6z^2 -8z + 8
since, ki is a root
f(ki)= z^4 - 2z3 + 6z^2 -8z + 8 = 0
substituting ki inside,
i get (k^2 -2)(k^2 -4) +2k(k^2 -4)i =0 Notice that this factorises as $\color{red}(k^2-4)(k^2-2+2ki)=0$.

im not sure how to proceed at this point..

the answer is z=2i, -2i, 1+i or 1-i
..