Long Division with Polynomials

Introduction

I think this is a very manual process and easy to get wrong but I guess it at least follows a pattern. I would how much of this is required now.

Example 1

So getting more tricky below we need to follow a process. The question is

(x² + 5x + 6) / (x + 2)

So we put the denominator on the left like normal long division

        -----------------------------
(x + 2) | x² + 5x + 6

Step 1: Next we divide (x² + 5x + 6) by the first term only

          x
        -----------------------------
(x + 2) | x² + 5x + 6

Step 2: Next we need to multiply everything in the denominator by the result (top row) x and reverse the sign

          x
        -----------------------------
(x + 2) | x² + 5x + 6
        -(x² - 2x)

Step 3: Now calculate the remainder by take one from the other

          x
        -----------------------------
(x + 2) | x² + 5x + 6
        -(x² - 2x)
        -----------------------------
          0  + 3x + 6

No next we do steps 1-3 again,

Step 1: So x / 3x = + 3

          x + 3 
        -----------------------------
(x + 2) | x² + 5x + 6
        -(x² - 2x)
        -----------------------------
          3x + 6

Step 2: Next we need to multiply everything in the denominator by the result (top row) x and reverse the sign

          x + 3 
        -----------------------------
(x + 2) | x² + 5x + 6
        -(x² - 2x)
        -----------------------------
          3x + 6
         -(3x +6)
          -------
               0

Step 3: Now calculate the remainder by take one from the other give 0

So we can say

 (x² + 5x + 6) / (x + 2) = x+3

Example 2

So here is the next and shows you what to do when you have something left over. Here is the question and workings

2x³+8x²-6x+10 / x-2

    2x²+12x+18+ 46/(x-2)
   -----------------------------

x-2 | 2x³+8x²-6x+10

   -(2x³-4x²)
   ---------------
         12x²-6x+10
       -(12x²-24x)
       -----------
              18x+10
             -(18x-36)
             ---------
                    46 

So the answer is 2x²+12x+18 + 46 / x-2 left over.

Example 3

So the final example was

6x⁴-9x²+18 / x-3

The trick to this question to fill in the missing terms first

6x⁴+0x³-9x²+0x¹+18 / x-3

Filling in these means you can apply exactly the same steps