1.

The HCF and LCM of polynomials is (a2 - b2) and (a2 – ab + b2) if one of the polynomial is (a - b) then find the other polynomial?1. (a + b) 3 - 3ab(a + b)2. (a - b) 3 - 3ab(a + b)3. (a + b) 3 + 3ab(a + b)4. (a + b) 3 - 3ab(a - b)

Answer» Correct Answer - Option 1 : (a + b) 3 - 3ab(a + b)

Given:

The HCF and LCM of polynomials is (a2 - b2) and (a2 – ab + b2) and one of the polynomial is (a - b)

Formula used:

Product of polynomial = Product of HCF and LCM of polynomials

p(x) × q(x) = LCM of [p(x) and q(x)] × HCF of [p(x) and q(x)]

Identity: (a + b) 3 = a3 + b3 + 3ab(a + b)

Identity: a2 - b2 = (a + b) × (a - b)

Identity: a3 + b3 = (a + b) × (a2 – ab + b2)

Calculation:

Let the other polynomial is q(a,b)

∴ q(a, b) = {LCM of (p(a, b) and q(a, b)) × HCF of (p(a,b) and q(a, b))}/p(a, b)

⇒ q(a, b) = {(a2 - b2) × (a2 – ab + b2)}/(a - b)

Now, we know the identify a2 - b2 = (a + b) × (a - b)

∴ q(a, b) = {(a + b) × (a - b) × (a2 – ab + b2)}/(a - b)

⇒ q(a, b) = {(a + b) × (a2 – ab + b2)}/(a - b)

⇒ q(a, b) = a3 + b3

This polynomial can also written in other form by using the identity

∴ q(a, b) = (a + b) 3 - 3ab(a + b)

Hence, option (1) is correct


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