HCF and LCM of two polynomials are x - 2 and x³ + 3x² - 6x - 8 respectively. If one polynomial is x² - x - 2, find the second polynomial?
- x² + 2x - 8
- x² - 2x - 8
- x² - 2x + 8
- None of these
Explanation
HCF = x - 2
LCM = x^3 + 3x^2 - 6x - 8
One polynomial = x^2 - x - 2 = (x - 2)(x + 1)
Factorize LCM:
x^3 + 3x^2 - 6x - 8 = (x - 2)(x^2 + 5x + 4) = (x - 2)(x + 1)(x + 4)
Using HCF × LCM = Product of polynomials:
(x - 2) × (x - 2)(x + 1)(x + 4) = (x - 2)(x + 1) × Second polynomial
Second polynomial = (x - 2)(x + 4) = x^2 + 2x - 8
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