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Factorise each of the following: (i) 8a^3 + b^3 + 12a^2b + 6ab^2 (ii) 8a^3 – b^3 – 12a^2b + 6ab^2

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Factorise each of the following:

(i) 8a3 + b3 + 12a2b + 6ab2

(ii) 8a3 – b3 – 12a2b + 6ab2

(iii) 27–125a3–135a +225a2

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(i) 8a3 + b3 + 12a2b + 6ab2

The expression, 8a3+b3+12a2b+6ab2 can be written as (2a)3+b3+3(2a)2b+3(2a)(b)2

8a3+b3+12a2b+6ab= (2a)3+b3+3(2a)2b+3(2a)(b)2

= (2a+b)3

= (2a+b)(2a+b)(2a+b)

Here, the identity, (x +y)3 = x3+y3+3xy(x+y) is used.

(ii) 8a3–b3–12a2b+6ab2

The expression, 8a3–b3−12a2b+6ab2 can be written as (2a)3–b3–3(2a)2b+3(2a)(b)2

8a3–b3−12a2b+6ab= (2a)3–b3–3(2a)2b+3(2a)(b)2

= (2a–b)3

= (2a–b)(2a–b)(2a–b)

Here, the identity,(x–y)3 = x3–y3–3xy(x–y) is used.

(iii) 27–125a3–135a+225a2 

The expression, 27–125a3–135a +225a2 can be written as 33–(5a)3–3(3)2(5a)+3(3)(5a)2

27–125a3–135a+225a=
33–(5a)3–3(3)2(5a)+3(3)(5a)2

= (3–5a)3

= (3–5a)(3–5a)(3–5a)

Here, the identity, (x–y)3 = x3–y3-3xy(x–y) is used.

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