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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 a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2} ) ]

= (a + b - (2a - 2b))((a + b)^{2} + (a + b)(2a - 2b) + (2a - 2b)^{2})

= (a + b - 2a + 2b)(a^{2} + b^{2} + 2ab + (a + b)(2a - 2b) + (2a - 2b)^{2}) [∵ (a + b)^{2} = a^{2} + b^{2} + 2ab]

= (3b - a)(a^{2} + b^{2} + 2ab + 2a^{2} - 2ab + 2ab - 2b^{2} + (2a - 2b)^{2})

= (3b - a)(3a^{2} + 2ab - b^{2} + (2a - 2b)^{2} )

= (3b - a)(3a^{2} + 2ab - b^{2} + 4a^{2} + 4b^{2} - 8ab) [∵ (a - b)^{2} = a^{2} + b^{2} - 2ab]

= (3b - a )(3a^{2} + 4a^{2} - b^{2} + 4b^{2} + 2ab - 8ab)

= (3b - a )(7a^{2} + 3b^{2} - 6ab)

∴ (a + b)^{3} - 8(a - b)^{3} = (-a + 3b)(7a^{2} - 6ab + 3b^{2})

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