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Factorise the following:

(i) x3 + 8y3 + 6x2y + 12xy2

(ii) 27a3 – 8b3 – 54a2b + 36ab2

(iii) 27 – 125x3 – 135x + 225x2

(iv) 125x3 + 64y3 – 300x2y + 240xy2

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(i) x3 + 8y3 + 6x2y + 12xy2

= (x)3 + (2y)3 + 3(x)(2y) [x + 2y]

= (x + 2y)3

[∵ (a + b)3 = a3 + b3 + 3 ab(a + b)]

(ii) 27a3 – 8b3 – 54a2b + 36ab2

= (3a)3 – (2b)3 – 3(3a)(2b) [3a – 2b]

= (3a – 2b)3

[∵ (a – b)3 = a3 – b3 – 3ab(a – b)]

(iii) 27 – 125x3 – 135x + 225x2

= (3)3 – (5x)3 – 3(3)(5x) [3 – 5x]

= (3 – 5x)3

[∵ (a – b)3 = a3 – b3 – 3ab(a – b)]

(iv) 125x3 – 64y3 – 300x2y + 240xy2

= (5x)3 – (4y)3 – 3(5x)(4y) [5x – 4y]

= (5x – 4y)3

[∵ (a – b) = a3 – b3 – 3ab(a – b)]

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