Question

A cube minus 27 b cube + 2 a square b minus 6 a b square

Answer 1

$\textbf{Given:}$

$a^3-27\,b^3+2\,a^2b-6\,ab^2$

$\textbf{To find:}$

$\text{Factors of a^3-27\,b^3+2\,a^2b-6\,ab^2 }$

$\textbf{Solution:}$

$\text{Consider,}$

$a^3-27\,b^3+2\,a^2b-6\,ab^2$

$=a^3-(3b)^3+2\,a^2b-6\,ab^2$

$=a^3-(3b)^3+2ab(a-3b)$

$\text{Using the following identity}$

$\boxed{\bf\,x^3-y^3=(x-y)(x^2+xy+y^2)}$

$=(a-3b)(a^2+3\,ab+9b^2)+2ab(a-3b)$

$=(a-3b)[a^2+3\,ab+9b^2+2ab]$

$=(a-3b)(a^2+5\,ab+9b^2)$

$\therefore\textbf{The factors of \bf\,a^3-27\,b^3+2\,a^2b-6\,ab^2 are \bf(a-3b) and \bf(a^2+5\,ab+9b^2) }$

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