Question

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 1

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