Given that k=2i is a root of f(x)= x^4 + 2x^3 + 8x^2 + 8x + 16, find the roots.

Answer 1

by Descartes sign rule, no + real roots

0,2 or 4 - real roots or imaginary roots

turns out all roots are imaginary, all 4

x = -1 +/-sqr3, +/-2i

graph the 4 degree polynomial, it never touches the x axis, so all roots are imaginarysqr3

imaginary roots come in conjugate pairs

x = +/-2i

(x+2i)(x-2r) = x^2 +4, divide that into the 4 degree polynomial to get x^2 +2x +4

use the quadratic formula or complete the square to get x = -1+/-sqr3