Question
Factorise $:a^3- 27b^3+ 2a^2b - 6ab^2$
✓
Answer
$a^3- 27b^3+ 2a^2b - 6ab^2$
We know that,
$a^3 - b^3 = ( a - b )( a^2 + ab + b^2 ) ....(1)$
$a^3 - 27b^3 + 2a^2b - 6ab^2$
$= (a)^3 - (3b)^3 + 2ab( a - 3b )$
$= ( a - 3b )[ a^2 + a \times 3b + (3b)^2 ] + 2ab( a - 3b ) [$From$(1)]$
$= ( a - 3b )[ a^2 + 3ab + 9b^2 ] + 2ab( a - 3b )$
$= ( a - 3b )[ a^2 + 3ab + 9b^2 + 2ab ]$
$= ( a - 3b )[ a^2 + 5ab + 9b^2 ]$
We know that,
$a^3 - b^3 = ( a - b )( a^2 + ab + b^2 ) ....(1)$
$a^3 - 27b^3 + 2a^2b - 6ab^2$
$= (a)^3 - (3b)^3 + 2ab( a - 3b )$
$= ( a - 3b )[ a^2 + a \times 3b + (3b)^2 ] + 2ab( a - 3b ) [$From$(1)]$
$= ( a - 3b )[ a^2 + 3ab + 9b^2 ] + 2ab( a - 3b )$
$= ( a - 3b )[ a^2 + 3ab + 9b^2 + 2ab ]$
$= ( a - 3b )[ a^2 + 5ab + 9b^2 ]$
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