Question

Factorize
i) $27m^3-216n^3$ ii) $343a^3-512b^3$

Answer 1

i) $27(m^3-8n^3)$
...[Taking out the common factor 27]
$=27[m^3-(2n{)}^3]$
Here, a = m and b = 2n
$\therefore 27m^3-216n^3$
$=27\left\{\begin{array}{c}(m-2n)[m^2+m(2n)+(2n{)}^2]\end{array}\right\}$
....$[\because a^3-b^3=(a-b\left)\right(a^2+ab+b^2\left)\right]$
$=27(m-2n)(m^2+2mn+4n^2)$

ii) $343a^3-512b^3$
$=(7a{)}^3-(8b{)}^3$
Here, A = 7a and B = 8b
$\therefore 343a^3-512b^3$
$=(7a-8b)\left[\right(7a{)}^2+\left(7a\right)\left(8b\right)+\left(8b{)}^2\right]$
...$[\because a^3-b^3=(a-b\left)\right(a^2+ab+b^2\left)\right]$
$=(7a-8b)(49a^2+56ab+64b^2$