1 / x² + x + 1 / x² - x
I am unsure as to how this is done. Any help is greatly appreciated.
Thank you.
$\displaystyle \frac{1}{x^2 + x} + \frac{1}{x^2 - x} = \frac{1}{x(x + 1)} + \frac{1}{x(x - 1)}$
so you should try to put over a common denominator of x(x+1)(x-1):
$\displaystyle = \frac{x - 1}{x(x + 1)(x-1)} + \frac{x+1}{x(x - 1)(x+1)}$
$\displaystyle = \frac{(x - 1) + (x + 1)}{x(x + 1)(x-1)}$
$\displaystyle = \frac{2x}{x(x + 1)(x-1)}$
$\displaystyle = \frac{2}{(x + 1)(x-1)}, \, x \neq 0$
$\displaystyle = \frac{2}{x^2 - 1}$.
You can check this by plugging in a value or two of x and seeing that you get the same answer for each expression.