For the following exercises, use synthetic division to determine the quotient involving a complex number.
Problem π₯ + 1 / π₯ + π
Answer 1 + 1 β π / π₯ + π
I get the correct answer when I do it via long division but I don't get the same answer for synthetic.
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$\begingroup$Long division:$$\require{enclose}
\begin{array}{rll} 1 \phantom{000}\\[-3pt] x+i \enclose{longdiv}{\quad x+1\phantom{000}}\kern-.2ex \\[-3pt] \underline{x+i\phantom{000}} \\[-3pt] 1-i \\[-3pt] \end{array}$$Synthetic division:$$\require{enclose}
\begin{array}{rll} -i\;\; \enclose{longdiv}{\quad1\qquad1\phantom{000}}\kern-.2ex \\[-3pt] \underline{\phantom{0000}-i\phantom{000}} \\[-3pt] 1\qquad 1-i \\[-3pt] \end{array}$$In both cases you end up with: quotient $\;1\;$ and remainder $\;1-i\;$
Here is a good summary polynomial synthetic division.
This is what my synthetic division looks like. What does your look like?
$\begin{array}{c|cc} &1&1\\-i&&-i\\\hline&1&1-i\end{array}$
Otherwise, I would say
$\frac {x+1}{x+i} = \frac {(x-i) -i + 1}{x+i} = 1 + \frac {1-i}{x+i}$
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