LOPANA STHÂPANÂBHYÂM

Lopana sthapanabhyam means 'by alternate elimination and retention'.

Consider the case of factorization of quadratic equation of type ax² + by² + cz² + dxy + eyz + fzx. This is a homogeneous equation of second degree in three variables x, y, z. The sub-sutra removes the difficulty and makes the factorization simple. The steps are as follows:

i) Eliminate z by putting z = 0 and retain x and y and factorize thus obtained a quadratic in x and y by means of Adyamadyena sutra.

ii) Similarly eliminate y and retain x and z and factorize the quadratic in x and z.

iii) With these two sets of factors, fill in the gaps caused by the elimination process of z and y respectively. This gives actual factors of the expression.

Example 1 : 3x² + 7xy + 2y² + 11xz + 7yz + 6z².

Step (i)  :   Eliminate z and retain x, y; factorize
              3x2 + 7xy + 2y2  = (3x + y) (x + 2y)
Step (ii) :   Eliminate y and retain x, z; factorize
              3x2 + 11xz + 6z2 = (3x + 2z) (x + 3z)
Step (iii):   Fill the gaps, the given expression
                               = (3x + y + 2z) (x + 2y + 3z)

Example 2 : 3x² + 6y² + 2z² + 11xy + 7yz + 6xz + 19x + 22y + 13z + 20

Step (i)  :   Eliminate y and z, retain x and independent term
              i.e., y = 0, z = 0 in the expression (E).
              Then E = 3x2 + 19x + 20 = (x + 5) (3x + 4)
Step (ii) :   Eliminate z and x, retain y and independent term
              i.e., z = 0, x = 0 in the expression.
              Then E = 6y2 + 22y + 20 = (2y + 4) (3y + 5)
Step (iii):   Eliminate x and y, retain z and independent term
              i.e., x = 0, y = 0 in the expression.
              Then E = 2z2 + 13z + 20 = (z + 4) (2z + 5)
Step (iv) :   The expression has the factors 
              (think of independent terms: constants)
                     = (3x + 2y + z + 4) (x + 3y + 2z + 5).

Solve the following expressions into factors by using appropriate sutras:

1. x² + 2y² + 3xy + 2xz + 3yz + z².

2. 3x² + y² - 4xy - yz - 2z² - zx.