We want to evaluate
(x⁵ + 15x⁴ + 54x³ - 25x² - 75x - 42)/(x+8)
There are two ways to do it: (a) long division, and (b) synthetic division.
Long Division
x⁴ + 7x³ - 2x² - 9x - 3
-------------------------------------------------
x+8 | x⁵ + 15x⁴ + 54x³ - 25x² - 75x - 42
x⁵ + 8x⁴
------------------------------------------------
7x⁴ + 54x³ - 25x² - 75x - 42
7x⁴ + 56x³
---------------------------------------
- 2x³ - 25x² - 75x - 42
- 2x³ - 16x²
---------------------------------
- 9x² - 75x - 42
- 9x² - 72x
----------------------------
- 3x - 42
- 3x - 24
-----------------
- 18
Answer:
x⁴ + 7x³ - 2x² - 9x - 3 - 18/(x+8)
Synthetic Division
--------------------------------------
-8 | 1 15 54 -25 -75 -42
-8 -56 16 72 24
-----------------------------------
1 7 -2 -9 -3 -18
Answer:
x⁴ + 7x³ - 2x² - 9x - 3 - 18/(x+8)