1. Vedic Math
The ancient Chinese method is guessing the factorized cofficients :
X Y Z
1 -3 2
1 4 -5
Below column X:
(1x)(1x)= 1x²
Below Y :
(-3y)(4y)=-12y²
Below Z:
(2z)(-5z)=-10z²
Check XY :
(1x4y)+ (-3y1x)=1 xy
Check YZ:
(2z4y)+(-3y)(-5z)= 23yz
Check XZ:
(-5z1x)+(2z1x)= -3xz
Answer
= (x-3y+2z)(x+4y-5z)
中学代数：增减辗转相除法
to find the remainder of f(x) when x= u
If f(u)=0,
then x=u is a root of f(x)
or
(x-u) a factor of f(x).
In Abstract Algebra (Ring Theory since Polynomial has Ring structure behaves exactly like Integers),
we note f(x)/(x-u)
where
(x-u) is the IDEAL of f(x).
Theory:
f(x)=p(x).(x-u)+r(x) …(1)
At x= u, (x-u)=0
f(u)= r(u)
r(u) being the remainder.
If r(u)=0,
from (1):
f(u)=p(u).(x-u)
then (x-u) is a factor (IDEAL) , or
x=u is a root of f(x).
Algebraic Geometry is a study of all IDEALS of the polynomials f(x). Like study D24/猫山/黑刺 durians, just enough by analysing their kernel (核)。
Note: Idéal in “Ring” is similar to Kernel in “Group”.
They are both the “essence” (aka “DNA”) of the structure Ring or Group, respectively.
Lisper 