Find the quotient using synthetic division with imaginary number $\frac{x+1}{x-i}$

I am to find the quotient using synthetic division: $\frac{x+1}{x-i}$

The solution is provided as $1+\frac{1+i}{x-i}$

I get $2+i$.

My working:

$$\begin{array} & i & | & 1 & 1 \end{array}$$

1. Pull down the 1 then multiply by i

2. sum 1 and i

$1+(1+i)$ = $2+i$

How can I arrive at $1+\frac{1+i}{x-i}$?

1 Answer

Your work is already correct (except for the step where you did $1+(1+i)$). Remember that the $(1+i)$ at the end of the synthetic division is the remainder, so what you ended up with is indeed $1+\frac{1+i}{x-i}$. You should not have added $(1+i)$ and $1$ together.