Factorise x^2+1 x^2 +2-3 x-3 x . - Vidyadip

Question

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

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Answer

$\begin{array}{l}\text { We have, } x^2+\frac{1}{x^2}+2-3 x-\frac{3}{x} \\ =(x)^2+\left(\frac{1}{x}\right)^2+2(x) \times\left(\frac{1}{x}\right)-3 x-\frac{3}{x} \\ \quad\left[\text { since, } 1=x \times \frac{1}{x}\right]\end{array}$
$\begin{array}{l}=\left(x+\frac{1}{x}\right)^2-3\left(x+\frac{1}{x}\right) \\ \quad\left[\text { using Identity I, where } a=x, b=\frac{1}{x}\right] \\ =\left(x+\frac{1}{x}\right)\left[\left(x+\frac{1}{x}\right)-3\right]=\left(x+\frac{1}{x}\right)\left(x+\frac{1}{x}-3\right)\end{array}$

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