Factorize: (a + b)^3 – 8(a – b)^3.
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The expression is in the form of a³ - b³
(a +b)³ - (2a - 2b)³ = [ a + b - (2a -2b) ] [ (a+b)² + (a+b)2(a-b)+(2a-2b)² ]
= (3 b - a) [ a² + b² + 2 a b + 2 a² - 2b² + 4a² + 4b² -8ab ]
= (3 b - a) (7 a² - 3 b² - 6 a b ) --- step 1
= (3 b - a) 7 [ a - b (3 +√30)/7 ] [ a - b (3 -√30)/7 ]
The factors of the second factor on the RHS in step 1, can be found using quadratic equation equating it 0.
(a +b)³ - (2a - 2b)³ = [ a + b - (2a -2b) ] [ (a+b)² + (a+b)2(a-b)+(2a-2b)² ]
= (3 b - a) [ a² + b² + 2 a b + 2 a² - 2b² + 4a² + 4b² -8ab ]
= (3 b - a) (7 a² - 3 b² - 6 a b ) --- step 1
= (3 b - a) 7 [ a - b (3 +√30)/7 ] [ a - b (3 -√30)/7 ]
The factors of the second factor on the RHS in step 1, can be found using quadratic equation equating it 0.
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