The Remainder Theorem

Introduction

No idea where this is going but here goes. They start with an example. Asked the robot what is was and it said

If you divide a polynomial f(x) by (x-a) the remainder is simply f(a) 

Example 1

In this we appear to use the approach of substitution and synthetic division

f(x) = 2x³ - 5x² + 6x -12

We are asked to workout when f(4). Using substitution we get

f(4) = 2(4)³ - 5(4)² + 6(4) -12
     = 128 - 80 + 24 - 12
f(4) = 60

They went of to use synthetic division so write down the coefficient and bring down the 2

4 | 2 - 5 + 6 - 12
  | ↓
  -----------------
    2

Continue Multiplying and adding

4 | 2 - 5 +  6 - 12
  | ↓ + 8 + 12 + 72
  -----------------
    2x²+3x + 18 = 60

Example 2

Here we go again

f(x) = 3x⁴ - 7x³ + 0x² - 9x +12 where f(5)
     = 3(5)⁴ -7(5)³   + 0 -9(5) + 12
     = 3(625) -7(125) - 45 + 12
     = 1875 - 875 -33
     = 1000 - 33
     =  967  

5 | 3 - 7    0  - 9 +   12
  | ↓  15   40  200 +  955 
  ------------------------
    3   8   40  191  = 967
    3x³+8x²+40x+191  = 967        

So

• the divisor is x − 5, so a = 5
• the polynomial is f(x) = 3x⁴ − 7x³ + 0x² − 9x + 12
• the computed f(5) = 967
• the synthetic division remainder = 967

So the theorem becomes:
If you divide the polynomial 3x⁴ − 7x³ + 0x² − 9x + 12 by (x-5),
the remainder is simply f(5)=967.

And that matches your synthetic division exactly.