Long Division with imaginary $i$

Question

The problem asks for the student to divide $(ix^4 + 3x^3 -ix^2 + ix + 4i +6)$ by $(x-2i)$ using both synthetic division as well as long division. When using synthetic division it seems that I MUST group the $4i+6$ together in order to get the correct answer and I don't understand why I need to do that.

Additionally, when I do the long division it seems that to get the same answer as the synthetic, I must also group the last 2 terms.
Is there a math reason for this? Why won't I get a valid answer by non-grouping? I get a different answer when I don't group those last 2 terms.

Answer 1

The reason is that $4i+6$ is the constant term. Both the $4i$ and the $6$ together make up the constant term. Even though there is a real part and an imaginary part, these two parts must be taken together because the constant term of the polynomial consists of both of them.

Treating them separately would be the same as, for example, treating $5$ as $2+3$ if you were dividing $x^2 + 2x + 5$ by $x-1$.

This isn't specific to the constant term also. For example, you'd have to do something similar if instead of just $+ix$ you also had, say, $+ix + 2x$. Then you'd have to treat that as $(2+i)x$ and keep the $2$ and the $i$ together.