The LCM and HCF of polynomials is (x2 – y2) and (x2 + xy + y2). If one of the polynomial is (x + y) then, what will be the other polynomial?

The LCM and HCF of polynomials is (x2 – y2) and (x2 + xy + y2). If one of the polynomial is (x + y) then, what will be the other polynomial? Correct Answer x³ – y³

Given:

The LCM and HCF of polynomials is (x² – y²) and (x² + xy + y²)

One of the polynomial is (x + y)

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: a² – b² = (a + b) × (a - b)

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

Calculation:

Let the other polynomial is q (x, y)

∴ q(x, y) = {LCM of (p(x, y) and q (x, y)) × HCF of (p(x, y) and q(x, y))}/p(x, y)

⇒ q(x, y) = {(x² – y²) × (x² + xy + y²)}/(x + y)

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

∴ q(x, y) = {(x + y) × (x – y) × (x² + xy + y²)}/(x + y)

⇒ q(x, y) = (x – y) × (x² + xy + y²) = x³ – y³

Hence, option (1) is correct