Question

Factorize each of the following expressions:

$\frac{1}{27}{x}^3-y^3+125z^3+5xyz$

Answer 1

Given: $\frac{1}{27}{x}^3-y^3+125z^3+5xyz$

$\therefore \frac{1}{27}{x}^3-y^3+125z^3+5xyz$

$={\left(\frac{1}{3}x\right)}^3+(-y{)}^3+(5z{)}^3-3\times \left(\frac{1}{3}x\right)\times (-y)\times 5z$

$[\because a^3+b^3+c^3-3abc=(a+b+c\left)\right(a^2+b^2+c^2$

$-ab-bc-ca\left)\right]$

$=(\frac{x}{3}-y+5z)(\frac{x^{(}2)}{9}+y^2+25z^2+\frac{xy}{3}+5yz-\frac{5xz}{3})$

Hence, $\frac{1}{27}{x}^3-y^3+125z^3+5xyz$

$=(\frac{x}{3}-y+5z)(\frac{x^2}{9}+y^2+25z^2+\frac{xy}{3}+5yz-\frac{5xz}{3}).$