Question

The HCF of $x^2-\frac{1}{x^2},x^2-2+\frac{1}{x^2}$, and $x^3-\frac{1}{x^3}-3x-\frac{3}{x}$ is

• A. $x^2-\frac{1}{x^2}$
• B. ${(x-\frac{1}{x})}^3(x+\frac{1}{x})$
• C. $x-\frac{1}{x}$
• D. ${(x-\frac{1}{x})}^2$

Answer 1

Answer: (C)

$x-\frac{1}{x}$

$x^2-\frac{1}{x^2}=(x+\frac{1}{x})(x-\frac{1}{x})$
$x^2-2+\frac{1}{x^2}=\left(x^2\right)-2\left(x\right)\left(\frac{1}{x}\right)+{\left(\frac{1}{x}\right)}^2={(x-\frac{1}{x})}^2$
$x^3-\frac{1}{x^3}-3x+\frac{3}{x}=(x{)}^3-{\left(\frac{1}{x}\right)}^3-3(x{)}^2\times \left(\frac{1}{x}\right)+3\left(x\right){\left(\frac{1}{x}\right)}^2={(x-\frac{1}{x})}^3$
$HCF=(x-\frac{1}{x})$