### LOPANA STHÂPANÂBHYÂM

Lopana sthapanabhyam means 'by alternate elimination and retention'.

Consider the case of factorization of quadratic equation of type ax^{2} + by^{2} + cz^{2} + 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 :** 3x^{2} + 7xy + 2y^{2} + 11xz + 7yz + 6z^{2}.

Step (i) : Eliminate z and retain x, y; factorize

3x^{2}+ 7xy + 2y^{2}= (3x + y) (x + 2y)

Step (ii) : Eliminate y and retain x, z; factorize

3x^{2}+ 11xz + 6z^{2}= (3x + 2z) (x + 3z)

Step (iii): Fill the gaps, the given expression

= (3x + y + 2z) (x + 2y + 3z)

**Example 2 :** 3x^{2} + 6y^{2} + 2z^{2} + 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 = 3x^{2}+ 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 = 6y^{2}+ 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 = 2z^{2}+ 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. x

^{2}+ 2y

^{2}+ 3xy + 2xz + 3yz + z

^{2}.

2. 3x

^{2}+ y

^{2}- 4xy - yz - 2z

^{2}- zx.

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