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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: http://www.purplemath.com/modules/synthcmp2.htm