Technique explanation
This lesson introduces the sub-sutra Lopanasthapanabhyam (By Elimination and Retention). It provides a systematic method for factorising homogeneous quadratics in three or more variables by temporarily eliminating variables.
To factorise a complex expression like 2x² + 6y² + 3z² + 7xy + 11yz + 7zx, follow a three-step process: First, set z = 0 and factorise the remaining x, y terms. Second, set y = 0 and factorise the x, z terms. Third, splice the results by matching the common variable terms (usually x). Finally, verify the coefficient of the unused term (the yz term).
This method applies to homogeneous quadratics where every term has the same total degree. If the variables don't match perfectly during splicing, you must scale the factorisations accordingly.
Standard fallback: The method of undetermined coefficients or grouping terms by degree and variable, applying the AC method or quadratic formula to each variable subset independently to find factors.
Worked examples
Example 1: Factorise 2x² + 6y² + 3z² + 7xy + 11yz + 7zx
- Step 1 (z=0): 2x² + 7xy + 6y² factors into (2x + 3y)(x + 2y).
- Step 2 (y=0): 2x² + 7zx + 3z² factors into (2x + z)(x + 3z).
- Step 3 (Splice): Match the 2x and x terms from both sets to get (2x + 3y + z)(x + 2y + 3z).
- Check: Expand the factors (2x + 3y + z)(x + 2y + 3z) = 2x(x + 2y + 3z) + 3y(x + 2y + 3z) + z(x + 2y + 3z) = (2x² + 4xy + 6xz) + (3xy + 6y² + 9yz) + (xz + 2yz + 3z²) = 2x² + 6y² + 3z² + 7xy + 11yz + 7zx.
Example 2: Factorise 3x² + 5xy + 2y² + 4xz + 3yz + z²
- Step 1 (z=0): 3x² + 5xy + 2y² factors into (3x + 2y)(x + y).
- Step 2 (y=0): 3x² + 4xz + z² factors into (3x + z)(x + z).
- Step 3 (Splice): Match the 3x and x terms to get (3x + 2y + z)(x + y + z).
- Check: Expand the factors (3x + 2y + z)(x + y + z) = 3x(x + y + z) + 2y(x + y + z) + z(x + y + z) = (3x² + 3xy + 3xz) + (2xy + 2y² + 2yz) + (xz + yz + z²) = 3x² + 5xy + 2y² + 4xz + 3yz + z².
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