Vedic (Factorize)

Vedic Sutras:
[s1]: proportionally
[s2]: first by first and last by last

Example 1: E= 2x² + 7x +6

Split 7x = 3x+4x
First ratio of coefficient (2x²+3x) -> 2:3
Last ratio of coefficient (4x+6) -> 4:6=2:3
=> 1st factor = (2x+3)

2nd factor:
2x²/(2x) +6/(3)= (x+2)

=> E = (2x+3).(x+2)

Example 2: Factorize E(x, y, z) = x²+xy-2y²+2xz -5yz-3z²

1. Let z = 0
E'= x²+xy-2y² = (x+2y)(x-y)

2. Let y=0
E'= x²+2xz-3z² = (x+3z)(x-z)

=> E(x, y, z) = (x+2y+3z)(x-y-z)

Example 3:  P(x, y, z) = 3x² + 7xy + 2y² +11xz + 7yz + 6z² + 14x + 8y + 14z + 8

1. Eliminate y=z=0, retain x:

P = 3x²+14x+8= (x+4)(3x+2)

2. Eliminate x=z=0, retain y:

P = 2y²+8y+8 = (2y+4)(y+2)

3. Eliminate x=y=0, retain z:

P = 6z²+14z+8 =(3z+4)(2z+2)

=> P =(x+2y+3z+4).(3x+y+2z+2)