Question
Factorize: $a(a + b)^3 - 3a^2b(a + b)$
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Answer
$a(a + b)^3 - 3a^2b(a + b)$ Taking $(a + b)$ common in the two terms $= (a + b){a(a + b)^2 - 3a^2b}$
Now, using $(a + b)^2 = a^2 + b^2 + 2ab$
$= (a + b){a(a^2 + b^2 + 2ab) - 3a^2b}$
$= (a + b){a^3 + ab^2 + 2a^2b - 3a^2b}$
$= (a + b){a^3 + ab^2 - a^2b}$
$= (a + b)p{a^2 + b^2 - ab}$
$= p(a + b)(a^2 + b^2 - ab)$
$\therefore a(a + b)^3 - 3a^2b(a + b) = a(a + b)(a^2 + b^2 - ab)$
Now, using $(a + b)^2 = a^2 + b^2 + 2ab$
$= (a + b){a(a^2 + b^2 + 2ab) - 3a^2b}$
$= (a + b){a^3 + ab^2 + 2a^2b - 3a^2b}$
$= (a + b){a^3 + ab^2 - a^2b}$
$= (a + b)p{a^2 + b^2 - ab}$
$= p(a + b)(a^2 + b^2 - ab)$
$\therefore a(a + b)^3 - 3a^2b(a + b) = a(a + b)(a^2 + b^2 - ab)$
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