Question

verify x^3-y^3=(x-y)(x^2+xy+y^2)​

Answer 1

We know that,

(x - y)³ = x³ - y³ - 3xy(x - y)

Hence,

x³ - y³ = (x - y)³ + 3xy(x - y)

Take out x - y as common factor

→ x³ - y³ = (x - y)[(x - y)² + 3xy]

We know that,

(x - y)² = x² + y² - 2xy

Hence,

→ x³ - y³ = (x - y)(x² + y²- 2xy + 3xy)

→ x³ - y³ = (x - y)(x² + y² + xy)

Hence, verified.

Alternative Method:-

Let x = 2, y = 1

→ x³ - y³ = (x - y)(x² + y² + xy)

→ (2)³ - (1)³ = (2 - 1)[(2)² + (1)² + 2(1)]

→ 8 - 1 = (1)(4 + 1 + 2)

→ 7 = (1)(7)

→ 7 = 7

Hence, verified.