How do I do synthetic division with imaginary numbers?

Synthetic division with complex numbers is very similar to regular synthetic division.

For example, if you were dividing a polynomial by x + 2, you would use the value of x = 2 in synthetic division. If you were dividing by x + 2 + i, you would use -2 – i as the value.

The coefficients will work as normal. I have added a link to a video which shows an example step by step. It also shows an alternate method which works better than synthetic division for certain cases.

A helpful video on Polynomial division with complex numbers:

Purple math is an awesome source, in this case it does have an example with imaginary numbers on this website: