Technique explanation
The Highest Common Factor (HCF) of two polynomials must also divide their difference, according to the sutra Sankalana-Vyavakalanabhyam ("By addition and by subtraction"). By subtracting one polynomial from another to eliminate the highest degree term, we obtain a lower-degree expression that contains the HCF. Factoring this result and verifying the factors using the factor theorem identifies the HCF efficiently.
Standard fallback: The Euclidean Algorithm for polynomials, which involves repeated division and taking remainders until a remainder of zero is reached.
Worked examples
Example 1: Find HCF of x³ + 6x² + 11x + 6 and x³ - x² - 10x - 8
1. Subtract: (x³ + 6x² + 11x + 6) - (x³ - x² - 10x - 8) = 7x² + 21x + 14.
2. Simplify: Divide by 7 to get x² + 3x + 2.
3. Factor: (x + 1)(x + 2). Both roots satisfy the originals.
Result: x² + 3x + 2
Check: Verify x² + 3x + 2 divides the first polynomial: (x² + 3x + 2)(x + 3) = x³ + 3x² + 2x + 3x² + 9x + 6 = x³ + 6x² + 11x + 6. Match confirmed.
Example 2: Find HCF of x³ + 7x² + 14x + 8 and x³ + 8x² + 19x + 12
1. Subtract: (x³ + 8x² + 19x + 12) - (x³ + 7x² + 14x + 8) = x² + 5x + 4.
2. Factor: (x + 1)(x + 4). Testing x = -1 and x = -4 confirms they are roots.
Result: x² + 5x + 4
Check: Verify x² + 5x + 4 divides the second polynomial: (x² + 5x + 4)(x + 3) = x³ + 5x² + 4x + 3x² + 15x + 12 = x³ + 8x² + 19x + 12. Match confirmed.
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