Question

Use the formula to find the factors.

(8) $x^3-\frac{1}{x^3}$

Answer 1

$x^3-\frac{1}{x^3}$ can be factorised as:
$$\begin{gathered} x^3-\frac{1}{x^3} \\ ={\left(x\right)}^3-{\left(\frac{1}{x}\right)}^3 \\ =\left(x-\frac{1}{x}\right)\left[{\left(x\right)}^2+x\times \frac{1}{x}+{\left(\frac{1}{x}\right)}^2\right]\left[\mathrm{using}\mathrm{the}\mathrm{formula}{a}^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)\right] \\ =\left(x-\frac{1}{x}\right)\left(x^2+1+\frac{1}{x^2}\right) \end{gathered}$$