Complex Synthetic Division ©2006, Bob Beeman

This version is temporary, and will probably change soon. In particular, I want to add scaling, so that you don't have to deal with messy rational roots that involve fractions like 1/3 or 1/7 that don't work well in decimal.

This page illustrates a modification to synthetic division to allow division of polynomials with complex coefficients by complex binomials. In order to do this, the simple multiplication operation must be changed to a "FOIL" (First Outer Inner Last) of the now complex coefficients. It should be remembered that the L term is the product of two imaginary factors, and therefore the sign must be switched to account for the fact that i² = -1. As a reminder of this the box containing this factor is tinted red, which perhaps suggests "negative" (at least if you are an accountant).

Complex Synthetic Division Calculator

Click on the divisor or equation coefficients in the top line to change.
Click the "Calculate" button to do synthetic division.

Trial Root (Divisor)		X²		X¹		X⁰
Real	Imag	Real	Imag	Real	Imag	Real	Imag
-1	-1	1	0	2	0	2	0
		:	:	-1	0	-1	1
		:	:	0	-1	-1	-1
		1	0	1	-1	0	0
		X¹		X⁰		Remainder

The default example above is the Complex Synthetic division of
x² + 2x + 2 divided by (x + 1 + i), which is one of the two conjugate roots.
The result is: (x + 1 - i), which is the other conjugate root.

Read More

The modified synthetic division setup looks like this:

Divisor		X²		X¹		X⁰
Real	Imag	Real	Imag	Real	Imag	Real	Imag
		:	:	F	O	F	O
		:	:	L	I	L	I
		Real	Imag	Real	Imag	Real	Imag
		X¹		X⁰		Remainder

Divisor		X²		X¹		X⁰
Real	Imag	Real	Imag	Real	Imag	Real	Imag
-1	-1	1	0	2	0	2	0
		:	:	-1	0	-1	1
		:	:	0	-1	-1	-1
		1	0	1	-1	0	0
		X¹		X⁰		Remainder

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