Technique explanation
Vedic Mathematics uses the principles of Gunitasamuccayah and Gunakasamuccayah to verify factorisations. By evaluating a polynomial and its factors at x = 1 (sum of coefficients) and x = -1 (alternating sum), we can quickly detect errors. If the product of the sums of the factors does not equal the sum of the coefficients of the original polynomial, the factorisation is incorrect.
Standard fallback: Direct expansion of the factors to see if the resulting polynomial matches the original expression exactly.
Worked examples
Example 1: Check x² + 7x + 12 = (x + 3)(x + 4)
1. At x = 1: Left side = 1 + 7 + 12 = 20. Right side = (1 + 3)(1 + 4) = 4 × 5 = 20. Check passes.
Result: Verified at x = 1.
Check: Expand (x + 3)(x + 4) = x(x + 4) + 3(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12. Match confirmed.
Example 2: Check x² - 5x + 6 = (x - 2)(x - 3)
1. At x = -1: Left side = (-1)² - 5(-1) + 6 = 1 + 5 + 6 = 12. Right side = (-1 - 2)(-1 - 3) = (-3) × (-4) = 12. Check passes.
Result: Verified at x = -1.
Check: Expand (x - 2)(x - 3) = x(x - 3) - 2(x - 3) = x² - 3x - 2x + 6 = x² - 5x + 6. Match confirmed.
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