Introduction

Needed to know this for Rational Zero Theorem so here goes

Example

So we start with

(x³ -2x² -5x + 6) /  (x-3)

So to begin we list the coefficients of the functions in a row

| 1 - 2 - 5 +  6
|
-----------------

Next we solve the x-3, i.e x -3 = 0, therefore

x = 3

So first we bring down the 1 3 | 1 - 2 - 5 + 6

 | ↓
 -----------------
   1

Then we multiply the 3 by the number brought down (1) and place it under the next to the next coefficient 3 | 1 - 2 - 5 + 6

 | ↓   3
 -----------------
   1

Add the coefficient to the result 3, i.e. -2 + 3 = 1 3 | 1 - 2 - 5 + 6

 | ↓   3
 -----------------
   1   1

Then we multiply the 3 by the number brought down (1) 3 | 1 - 2 - 5 + 6

 | ↓   3   3 
 -----------------
   1   1

Add the coefficient to the result 3, i.e. -5 + 3 = -2 3 | 1 - 2 - 5 + 6

 | ↓   3   3 
 -----------------
   1   1  -2

Then we multiply the 3 by the number brought down (-2) 3 | 1 - 2 - 5 + 6

 | ↓   3   3 - 6
 -----------------
   1   1  -2

Add the coefficient to the result 3, i.e. 6 - 6 = 0 3 | 1 - 2 - 5 + 6

 | ↓   3   3 - 6
 -----------------
   1   1  -2   0

If the you get a zero then

 x - 3 is a factor and x =3

So taking the last line and adding the power of x in line with the question we see

 -----------------
   1   1  -2   0
   1x³+1x -2 <= This is the answer