Let’s factorise :

i.

ii.

iii.

iv.

v.

vi.

vii.

viii.

ix.

x.

xi.

xii.

xiii.

_{i. 225m}² – 100n²

= (15m)² – (10n)²

[∵ a² – b² = (a – b)(a + b)]

Here, a = 15m, b = 10n

= (15m – 10n)(15m + 10n)

ii.

[∵ a² – b² = (a – b)(a + b)]

Here, a = 5x,

iii. 7ax² + 14ax + 7a

= 7a(x² + 2x + 1)

Since, a² + 2ab + b²= (a + b)²

Here, a = x, b = 1

= 7a(x + 1)²

iv. 3x⁴ – 6x²a² + 3a⁴

= 3(x⁴ – 2x²a² + a⁴)

= 3[(x²)² – 2(x²)(a²) + (a²)²]

Since, a² + 2ab + b²= (a + b)²

Here, a = x², b = a²

= 3(x² – a²)²

[∵ a² – b² = (a – b)(a + b)]

Here, a = x, b = a

= 3[(x – a)(x + a)]²

= 3(x – a)²(x + a)²

v. 4b²c² – (b² + c² – a²)²

= (2bc)² – (b² + c² – a²)²

= (2bc + b² + c² – a²) (2bc – (b² + c² – a²)) [∵ a² – b² = (a – b)(a + b)]

= (b² + c² + 2bc – a²) (2bc – b² – c² + a²)

[∵ (a + b)² = a² + 2ab + b²]

So, if a = b, b = c in b² + c² + 2bc then after applying the identity we get,

b² + c² + 2bc = (b + c)²

= [(b + c)² – a²] [a² – (b² + c² – 2bc)]

= (b + c – a) (b + c + a)[a²– (b – c)²]

[∵ (a – b)² = a² – 2ab + b²_, Here a = b, b = c]

= (b + c – a) (b + c + a)(a – (b – c))(a + b – c)

= (b + c – a) (b + c + a)(a – b + c)(a + b – c)

vi. 64ax² – 49a(x – 2y)²

= a(64x² – 49(x – 2y)²)

= a[(8x)² – (7(x – 2y))²]

[∵ a² – b² = (a – b)(a + b)]

Here, a = 8x, b = 7(x – 2y)

= a[8x + 7(x – 2y)][8x – 7(x – 2y)]

= a(8x + 7x – 14y)(8x - 7x + 14y)

= a(15x – 2y)(x + 14y)

vii. x² – 9 – 4xy + 4y²

= x² – 4xy + 4y² – 9

= x² – 2(x)(2y) + (2y)² – 9

Since, a² - 2ab + b²= (a - b)²

Here, a = x, b = 2y

= (x – 2y)² - 3²

[∵ a² – b² = (a – b)(a + b)]

Here, a = x – 2y, b = 3

= (x – 2y + 3)(x – 2y – 3) [∵ a² – b² = (a – b)(a + b)]

viii. x² – 2x – y² + 2y

= x² – y² – 2x + 2y

= (x + y)(x – y) – 2(x – y)

[∵ a² – b² = (a – b)(a + b), here a = x, b = y]

= (x – y)(x + y – 2) [Taking (x – y) as common]

ix. 3 + 2a – a²

= 2 + 1 + 2a – a²

= 2 + 2a + 1² – a²

= 2(1 + a) + (1 – a)(1 + a)

[∵ a² – b² = (a – b)(a + b), here a = 1, b = a]

= (1 + a)(2 + 1 – a) [Taking (1 + a) as common]

= (1 + a)(3 – a)

x. x⁴ – 1

= (x²)² - 1²

= (x² – 1)(x² + 1)

[∵ a² – b² = (a – b)(a + b), here a = x², b = 1]

= (x – 1)(x + 1)(x² + 1)

[∵ a² – b² = (a – b)(a + b), here a = x, b = 1]

xi. a²– b² – c² + 2bc

= a² – (b² + c² – 2bc)

= a² – (b - c)²

[∵ (a – b)² = a² – 2ab + b², here a = b and b = c]

= (a + b – c)(a – (b – c))

[∵ a² – b² = (a – b)(a + b), here a = a, b = b - c]

= (a + b – c)(a – b +c)

xii. ac + bc + a + b

= ac + a + bc + b

= a(c + 1) + b(c + 1)

= (c + 1)(a + b) [Taking (c + 1) as common]

xiii. x⁴ + x²y² + y⁴

= x⁴ + 2x²y² – x²y² + y⁴

= x⁴ + 2x²y² + y⁴ – x²y²

= (x²)² + 2(x²)(y²) + (y²)² – (xy)²

= (x² + y²)² – (xy)²

[∵ (a + b)² = a² + 2ab + b², here a = x², b = y²]

= (x² + y² + xy)(x² + y² – xy)

[∵ a² – b² = (a – b)(a + b), here a = x² + y² and b = xy]