Question

How do you long divide `(20x^3 - 9x^2 + 18x + 12) / (4x+3)`?

Answer 1

There are various ways of writing the details Here's the one that I learned and teach (using a standard textbook in the US):

Explanation:

To divide: `(20x^3-9x^2 +18x + 12)/(4x+3)`

`" " " "" "` `--------`
`4x+3 )` `20x^3` `-9x^2` `+18x` `+12`

What do we need to multiply `4x` by, to get `20x^3`? We need to multiply by `5x^2`. Write that `5x^2` on the top line:

`" " " " " " "5x^2`
`" " " "" "` `--------`
`4x+3 )` `20x^3` `-9x^2` `+18x` `+12`

Now multiply `5x^2` times the divisor, `4x+3`, to get `20x^3+15x^2` and write that under the dividend.

`" " " " " " "5x^2`
`" " " "" "` `--------`
`4x+3 )` `20x^3` `-9x^2` `+18x` `+12`
`" "" "" "` `20x^3` `+15x^2`
`" " " " " "` `-----`

Now we need to subtract `20x^3+15x^2` from the dividend. (You may find it simpler to change the signs and add. I do, so I'll do it that way)

`" " " " " " "5x^2`
`" " " "" "` `--------`
`4x+3 )` `20x^3` `-9x^2` `+18x` `+12`
`" " " "` `color(red)(-)20x^3color(red)(-)15x^2`
`" "" "" "` `-----`
`" "" "" "" "" "` `-24x^2` `+18x` `+12`

Now, what do we need to multiply `4x` (the first term of the divisor) by to get `-24x^2` (the first term of the last line)? We need to multiply by `-6x`
So write `-6x` on the top line, then multiply `-6x` times the divisor `4x+3`, to get `-24x^2-18x` and write it underneath.

`" " " " " " "5x^2` `-6x`
`" " " "" "` `--------`
`4x+3 )` `20x^3` `-9x^2` `+18x` `+12`
`" " " "` `color(red)(-)20x^3color(red)(-)15x^2`
`" "" "" "` `-----`
`" "" "" "" "" "` `-24x^2` `+18x` `+12`
`" "" "" "" "" "` `-24x^2` `-18x`
`" " " "" "" "` `------`

Now subtract (change the signs and add), to get:

`" " " " " " "5x^2` `-6x`
`" " " "" "` `--------`
`4x+3 )` `20x^3` `-9x^2` `+18x` `+12`
`" " " "` `color(red)(-)20x^3color(red)(-)15x^2`
`" "" "" "` `-----`
`" "" "" "" "" "` `-24x^2` `+18x` `+12`
`" "" "" "" "" "` `color(red)(+)24x^2` `color(red)(+)18x`
`" " " "" "" "` `--------`
`" "" "" "" "" "" "" "" "" "` `36x` `+12`

We'll be done when the last line is `0` or has degree less than the degree of the divisor. Which has not happened yet, but we're close.

`9` times `4x` will get us `36x`, so we put the `9` on top multiply, subtract (change signs and add) to get:

`" " " " " " "5x^2` `-6x` `+9`
`" " " "" "` `--------`
`4x+3 )` `20x^3` `-9x^2` `+18x` `+12`
`" " " "` `color(red)(-)20x^3color(red)(-)15x^2`
`" "" "" "` `-----`
`" "" "" "" "" "` `-24x^2` `+18x` `+12`
`" "" "" "" "" "` `color(red)(+)24x^2` `color(red)(+)18x`
`" " " "" "" "` `--------`
`" "" "" "" "" "" "" "" "" "` `36x` `+12`
`" "" "" "" "" "" "" "" "` `color(red)(-)36x` `color(red)(-)27`
`" " " "" "" "" "" "` `--------`
`" "" "" "" "" "" "" "" "" "" "" "` `-15`

Now the last line has degree less than `1`, so we are finished.

The quotient is: `5x^2-6x+9` and the remainder is `-15`

We can write:

`(20x^3-9x^2 +18x + 12)/(4x+3) = 5x^2-6x+9 -15/(4x+3)`

IMPORTANT to understanding!
If we get a common denominator on the right and simplify we will get exactly the left side.