# Complex numbers problem

• Mar 31st 2010, 08:51 AM
mephisto50
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
• Mar 31st 2010, 12:09 PM
Opalg
Quote:

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 \$\displaystyle \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

..