Question

One of the factors of $x^2+\frac{1}{x^2}+2-2x-\frac{2}{x}$ is:

• 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 $x+\frac{1}{x}$

We can use the identity $a^2+2ab+b^2=(a-b{)}^2$ we can simplify the expression as follows.

$x^2+\frac{1}{x^2}+2-2x-\frac{2}{x}$
$=(x^2+\frac{1}{x^2}+2)-2(x+\frac{1}{x})$
$={(x+\frac{1}{x})}^2-2(x+\frac{1}{x})$
$=(x+\frac{1}{x})(x+\frac{1}{x}-2)$

Thus, the factors of $x^2+\frac{1}{x^2}+2-2x-\frac{2}{x}$ are
$(x+\frac{1}{x})$ and $(x+\frac{1}{x}-2)$.

Among these $(x+\frac{1}{x})$ is there in options.