Find : (a - b)(a - b)(a - b) - Vidyadip

Question

Find : $(a - b)(a - b)(a - b)$

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Answer

$(a + b)(a + b)(a + b)$
$= (a \times a + a \times b + b \times a + b \times b)(a + b)$
$= (a^2+ab+ab+ b^2)(a + b)$
$= (a^2+ b^2+ 2ab)(a + b)$
$= a^2\times a + a^2\times b + b^2\times a + b^2\times b + 2ab \times a + 2ab \times b$
$= a^3+ a^2b + ab^2+ b^3+ 2a^2b + 2ab^2$
$= a^3+ b^3+ 3a^2b + 3ab^2$
replacing b by $-b$, we get
$= a^3+ (-b)^3+ 3a^2(-b) + 3a(-b)^2$
$= a^3- b^3- 3a^2b + 3ab^2$

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