Question

Factorise: $a^3-\frac{1}{a^3}-2a+\frac{2}{a}$.

Answer 1

Factorize the polynomial

$$\begin{gathered} a^3-\frac{1}{a^3}-2a+\frac{2}{a} \\ ={\left(a\right)}^3-{\left(\frac{1}{a}\right)}^3-2\left(a-\frac{1}{a}\right) \end{gathered}$$

using identity $x^3-y^3=(x-y)\left(x^2+y^2+xy\right)$

$$\begin{gathered} =\left(a-\frac{1}{a}\right)\left(a^2+\frac{1}{a^2}+a\times \frac{1}{a}\right)-2\left(a-\frac{1}{a}\right) \\ =\left(a-\frac{1}{a}\right)\left(a^2+\frac{1}{a^2}+1-2\right) \\ =\left(a-\frac{1}{a}\right)\left(a^2+\frac{1}{a^2}-1\right) \end{gathered}$$

Hence, the factors of $a^3-\frac{1}{a^3}-2a+\frac{2}{a}$are $\left(a-\frac{1}{a}\right)$and $\left(a^2+\frac{1}{a^2}-1\right)$.