How would you use synthetic division to find the quotient and remainder for this problem:

Question

(X^3+3ix^2-4ix-2)/(x+i)

Answer 1

The same way you do it normally. Just keep track of the i's and remember that -i^2 = 1

So, the rows of coeffiicients are (munged by font spacing)

1 3i -4i -2 (i
0 i -4 -4i+4
_____________
1 4i -4-4i 2-4i

So that means that

(x^3 + 3ix^2 - 4ix - 2)/(x-i)
= x^2 + 4ix - (4+4i) remainder 2-4i

If you multiply it out, it works.