The premise
given factor (x−a) to f(x)=Bx2+Cx+D,
![enter image source here]()
The pink arrow means multiply a by whatever the difference of the previous column's value was (it's subtraction).
You start out with taking B and putting it in the difference row, so the next column will be C−aB. The next column should be D−a(C−aB), and so on. This applies to all polynomial equations.
Also, you CANNOT miss a term, so if you have something like 3x5+8x3−2x2 you would have to write in 0s for the missing terms...
like this →3x5+0x4+8x3−2x2+0x−0
Your result should have all the powers of x shifted down one degree, so a polynomial beginning with x5 would have a quotient beginning with x4.
If the last column does not get you a 0 as the result, then it will be the remainder, which you put over the factor you divided by, remainderx−a
Actual calculation
14x2−34↔14x2+0x−34
dividing by (x+4); from (x−a) form that means a=−4
So we have the synthetic division set up:
-4 | 14 0 -34
----.--------.------.
abc14
-4 | 14 a 0 abc-34
−abc(-56) a224
---.----------.----------.
abc14ab56abc258
We get 14x−56+258x−4
