Question
Factorize the following expressions: $\frac{1}{27}\text{x}^3-\text{y}^3+125\text{z}^3+5\text{xyz}$
✓
Answer
$\frac{1}{27}\text{x}^3-\text{y}^3+125\text{z}^3+5\text{xyz}$
$=\Big(\frac{\text{x}}{3}\Big)^3+(-\text{y})^3+(5\text{z})^3-3\times\frac{\text{x}}{3}(-\text{y})(5\text{z})$
$=\Big(\frac{\text{x}}{3}+(-\text{y})+5\text{z}\Big)\Big(\frac{\text{x}}{3}\Big)^2+(-\text{y})^2+(5\text{z})^2-\frac{\text{x}}{3}(-\text{y})-(-\text{y})5\text{z}-5\text{z}\Big(\frac{\text{x}}{3}\Big)\Big)$
$=\Big(\frac{\text{x}}{3}-\text{y}+5\text{z}\Big)\Big(\frac{\text{x}^2}{9}+\text{y}^2+25\text{z}^2+\frac{\text{xy}}{3}+5\text{yz}-\frac{5}{3}\text{zx}\Big)$
$\therefore\frac{1}{27}\text{x}^3-\text{y}^3+125\text{z}^3+5\text{xyz}$
$=\Big(\frac{\text{x}}{3}-\text{y}+5\text{z}\Big)\Big(\frac{\text{x}^2}{9}+\text{y}^2+25\text{z}^2+\frac{\text{xy}}{3}+5\text{yz}-\frac{5}{3}\text{zx}\Big)$
$=\Big(\frac{\text{x}}{3}\Big)^3+(-\text{y})^3+(5\text{z})^3-3\times\frac{\text{x}}{3}(-\text{y})(5\text{z})$
$=\Big(\frac{\text{x}}{3}+(-\text{y})+5\text{z}\Big)\Big(\frac{\text{x}}{3}\Big)^2+(-\text{y})^2+(5\text{z})^2-\frac{\text{x}}{3}(-\text{y})-(-\text{y})5\text{z}-5\text{z}\Big(\frac{\text{x}}{3}\Big)\Big)$
$=\Big(\frac{\text{x}}{3}-\text{y}+5\text{z}\Big)\Big(\frac{\text{x}^2}{9}+\text{y}^2+25\text{z}^2+\frac{\text{xy}}{3}+5\text{yz}-\frac{5}{3}\text{zx}\Big)$
$\therefore\frac{1}{27}\text{x}^3-\text{y}^3+125\text{z}^3+5\text{xyz}$
$=\Big(\frac{\text{x}}{3}-\text{y}+5\text{z}\Big)\Big(\frac{\text{x}^2}{9}+\text{y}^2+25\text{z}^2+\frac{\text{xy}}{3}+5\text{yz}-\frac{5}{3}\text{zx}\Big)$
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