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Factorise the following algebraic expressions by using the identity a2 – b2 = (a + b)(a – b).

(i) z2 – 16

(ii) 9 – 4y2

(iii) 25a2 – 49b2

(iv) x4 – y4

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(i) z2 – 16

z2 – 16 = z2 – 42

We have a2 – b2 = (a + b) (a – b)

Let a = z and b = 4,

z2 – 42 = (z + 4) (z – 4)

(ii) 9 – 4y2

9 – 4y2 = 32 – 22 y

= 32 – (2y)2

Let a = 3 and b = 2y, then

a2 – b2 = (a + b) (a – b)

∴ 32 – (2y)2 = (3 + 2y) (3 – 2y)

9 – 4y2 = (3 + 2y) (3 – 2y)

(iii) 25a2 – 49b2

25a2 – 49b= (5a)2 – (7b)2

Let A = 5a and B = 7b

AB2

(5a)2 – (7b)2 = (5a + 7b) (5a – 7b)

(iv) x4 – y4

Let x4 – y4 = (x2)2 – (y2)2

We have a2 – b2 = (a + b) (a – b)

(x2)2 – (y2)2 = (x2 + y2) (x2 – y2)

x4 – y4 = (x2 + y2) (x2 – y2)

Again we have x2 – y2 = (x + y) (x – y)

∴ x4 – y4 = (x2 + y2) (x + y) (x – y)

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