Factorize the Following Expressions: (A + B)3 – 8(A – B)3 - Mathematics

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Factorize the following expressions:

(a + b)3 – 8(a – b)3

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Solution

(a + b)3  - 8(a - b)3

= (a + b)3 - [2(a - b)]3

= (a + b)3 - (2a - 2b)3         [Using a3 - b3  = (a - b)(a2 + ab + b2 ) ]

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

= (a + b - 2a + 2b)(a2  + b2  + 2ab + (a + b)(2a - 2b) + (2a - 2b)2)    [∵ (a + b)2  = a2 + b2 + 2ab]

= (3b - a)(a2  + b2  + 2ab + 2a2  - 2ab + 2ab - 2b2  + (2a - 2b)2)

= (3b - a)(3a2  + 2ab - b2  + (2a - 2b)2 )

= (3b - a)(3a2 + 2ab - b2 + 4a2 + 4b2 - 8ab)       [∵ (a - b)2  = a2 + b2 - 2ab]

= (3b - a )(3a2 + 4a2 - b2 + 4b2 + 2ab - 8ab)

= (3b - a )(7a2 + 3b2 - 6ab)

∴ (a + b)3  - 8(a - b)3   = (-a + 3b)(7a2  - 6ab + 3b2)

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Chapter 5: Factorisation of Algebraic Expressions - Exercise 5.2 [Page 14]

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RD Sharma Mathematics for Class 9
Chapter 5 Factorisation of Algebraic Expressions
Exercise 5.2 | Q 17 | Page 14

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