Factorise

(i)

(ii)

(iii)

(iv)

(v) _{(l + m)}² - (l - m)²

(vi)

(vii)

(viii)

(i) 4p² - 9q²

4p² - 9q² = (2p)² - (3q)²

Using identity a² - b² = (a + b)(a - b)

Here a = 2p; b = 3q

(2p)² - (3q)² = (2p + 3q) (2p – 3q)

(ii) 63a² - 112b²

7(9a² - 16b²) = 7{(3a)² - (4b²)}

Using identity a² - b² = (a + b)(a - b)

Here a = 3a; b = 4b

7{(3a)² - (4b)²} = 7{(3a + 4b) (3a – 4b)}

(iii) 49x² - 36 = (7x)² - 6²

Using identity a² - b² = (a + b)(a - b)

Here a = 7x; b = 6

(7x)² - (6)² = (7x + 6) (7x – 6)

(iv) 16x⁵ - 144x³ = 16x³(x² - 9) ⇒ 16x³{x² - 3²}

Using identity a² - b² = (a + b)(a - b)

Here a = x; b = 3

16x³{(x)² - (3)²} = 16x³{(x + 3) (x – 3)}

(v) (l + m)² - (l - m)²

Using identity a² - b² = (a + b)(a - b)

Here a = (l + m); b = (l - m)

(l + m)² - (l + m)² = (l + m + l - m) (l + m – l + m)

⇒ (l + m)² - (l + m)² = 2l × 2m = 4lm

(vi) 9x²y² - 16 = (3xy)² - 4²

Using identity a² - b² = (a + b)(a - b)

Here a = 3xy; b = 4

(3xy)² - 4² = (3xy + 4) + (3xy - 4)

(vii) (x²- 2xy + y² ) = (x + y)² [using identity (a + b)² = a² + 2ab + b²]

(x + y)² - z² =

Using identity a² - b² = (a + b)(a - b)

Here a = _{(x + y)}² ; b = z

(x + y)² - z² = (x + y + z) + (x + y - z)

(viii) 25a² - (4b² - 28bc + 49c²) ⇒ (5a)² - (2b – 7c )² [using identity (a - b)² = a² - 2ab + b²]

Using identity a² - b² = (a + b)(a - b)

Here a = _{5a ; b = 4b – 7c}

(5a)² - (2b – 7c )² = (5a + 2b – 7c) + (5a – 2b + 7c)