Question 288565
The division process is shown below in detail.


Original expression is:


{{{(z^3-z^2-4)/(z+i)}}}


First you divide {{{z}}} into {{{z^3}}} to get {{{z^2}}}.
Then you multiply {{{z+i}}} by {{{z^2}}} to get {{{z^3 + z^2i}}}.
Then you subtract {{{z^3 + z^2i}}} from {{{z^3 - z^2}}} to get {{{-z^2 - z^2i}}}.
Then you divide {{{z}}} into {{{-z^2}}} to get {{{-z}}}.
Then you multiply {{{z+i}}} by {{{-z}}} to get {{{-z^2 - zi}}}.
Then you subtract {{{-z^2 - zi}}} from {{{-z^2 - z^2i}}} to get {{{-z^2i + zi}}}.
Then you divide {{{z}}} into {{{-z^2i}}} to get {{{-zi}}}.
Then you multiply {{{z+i}}} by {{{-zi}}} to get {{{-z^2i - zi^2}}}.
Then you subtract {{{-z^2i - zi^2}}} from {{{-z^2i + zi}}} to get {{{zi + zi^2}}}.
Then you divide {{{z}}} into {{{zi}}} to get {{{i}}}.
Then you multiply {{{z+i}}} by {{{i}}} to get {{{zi + i^2}}}.
Then you subtract {{{zi + i^2}}} from {{{zi + zi^2}}} to get {{{zi^2 - i^2}}}.
Then you divide {{{z}}} into {{{zi^2}}} to get {{{i^2}}}.
Then you multiply {{{z+i}}} by {{{i^2}}} to get {{{zi^2 + i^3}}}.
Then you subtract {{{zi^2 + i^3}}} from {{{zi^2 - i^2}}} to get {{{-i^2 - i^3}}}.
Then you bring down the {{{-4}}} to get {{{-i^2 - i^3 - 4}}}.


Since {{{i^2}}} = {{{-1}}}, you can substitute in this last expression to get:


{{{-(-1) - (-1*i) - 4}}} which becomes:


{{{1 + i - 4}}} which becomes:


{{{i - 3}}}.


That's your remainder.


Your answer is:


{{{(z^3-z^2-4)/(z+i)}}} = {{{z^2 - z - zi + i + i^2}}} with a remainder of {{{i - 3}}}.


To prove that this is correct, you need to multiply the answer by {{{z+i}}} and then add the remainder back in to see if you can duplicate the original expression.


You do that in the following manner.


{{{z^2 - z - zi + i + i^2}}} * {{{z+i}}} equals:


{{{z^2 - z - zi + i + i^2}}} * {{{z}}} plus:
{{{z^2 - z - zi + i + i^2}}} * {{{i}}}.


First we multiply {{{z^2 - z - zi + i + i^2}}} * {{{z}}}.


That becomes:


{{{z^3 - z^2 - z^2i + zi + zi^2}}}.


Then we multiply {{{z^2 - z - zi + i + i^2}}} * {{{i}}}.


That becomes:


{{{z^2i - zi - zi^2 + i^2 + i^3}}}.


Then we add:


{{{z^3 - z^2 - z^2i + zi + zi^2}}} and {{{z^2i - zi - zi^2 + i^2 + i^3}}} together to get:


{{{z^3 - z^2 - z^2i + zi + zi^2 + z^2i - zi - zi^2 + i^2 + i^3}}}.


Then we combine like terms to get:


{{{z^3 - z^2 + i^2 + i^3}}}.


{{{z^2i}}} and {{{-z^2i}}} canceled out.
{{{zi}}} and {{{-zi}}} canceled out.
{{{zi^2}}} and {{{-zi^2}}} canceled out.


You are left with:


{{{z^3 - z^2 + i^2 + i^3}}}.


Since {{{i^2}}} = {{{-1}}}, you can substitute in this expression to get:


{{{z^3 - z^2 + i^2 + i^3}}} = {{{z^3 - z^2 - 1 - i}}}.


You now need to add the remainder of {{{i - 3}}} back in.


You get:


{{{z^3 - z^2 - 1 - i}}} plus {{{i - 3}}} equals:


{{{z^3 - z^2 - 1 - i + i - 3}}}.


Combine like results to get:


{{{z^3 - z^2 - 4}}}.


The {{{-i}}} and the {{{i}}} canceled out.


Since this is the same as the original expression you started with, your division is confirmed as being successfully concluded.