Question

Factorise the following expression x 2 + 2 x y + y 2 − z 2 .

Answer 1

Answer:

(x

2

−2xy+y

2

)−z

2

⇒[x(x−y)−y(x−y)]−z

2

⇒(x−y)(x−y)−z

2

⇒(x−y)

2

−z

2

⇒[(x−y)+z][(x−y)−z]

⇒(x−y+z)(x−y−z)

Answer 2

Question:-

➡ Factorise the following polynomial.

Answer:-

➡ The factorised form is (x + y + z)(x + y - z)

Solution:-

x² + 2xy + y² - z²

= x² + xy + xy + y² - z²

= x(x + y) + y(x + y) - z²

= (x + y)(x + y) - z² (you can directly write (x + y)²)

= (x + y)² - z²

= (x + y + z)(x + y - z)

Hence, the factorised form is (x + y + z)(x + y - z)

Identity used:-

➡ x² + 2xy + y² = (x + y)²

➡ x² - y² = (x + y)(x - y)

More Identities to Learn:-

➡ x² - 2xy + y² = (x - y)²

➡ (x + y)³ = x³ + y³ + 3xy(x + y)

➡ (x - y)³ = x³ - y³ - 3xy(x - y)