Question

Which among the following is one of the factors of $x^2+\frac{1}{x^2}+2-2x-\frac{2}{x}$?

• A. $x-\frac{1}{x}$
• B. $x+\frac{1}{x}-1$
• C. $x+\frac{1}{x}$
• D. $x^2+\frac{1}{x^2}$

Answer 1

The correct option is C $x+\frac{1}{x}$

$f\left(x\right)=x^2+\frac{1}{x^2}+2-2x-\frac{2}{x}$
$=x^2+\frac{1}{x^2}+2x.\frac{1}{x}-2[x+\frac{1}{x}]$
$={[x+\frac{1}{x}]}^2-2[x+\frac{1}{x}]$
$=[x+\frac{1}{x}][x+\frac{1}{x}-2]$
$\therefore [x+\frac{1}{x}]$ and $[x+\frac{1}{x}]$ are factors of $f\left(x\right)$