How does the fraction x^2/(x^2-1) simplify to 1 + 1/(x^2-1)?

[tony873004]

This is a small step in a larger Calc problem. There's 2 problems in a row where this same arrangement popped up. I have a feeling I'm forgetting something basic. How does {\frac{{x^2 }}{{x^2 - 1}}} become 1 + \frac{1}{{x^2 - 1}} ?

 

[Dick]

Use polynomial long division to divide the numerator by the denominator. This gives you quotient, 1, and a remainder, 1. Just like 5/4=1+1/4.

 

[stunner5000pt]

add and subtract 1 from the numerator

then we get


\frac{x^2}{x^2 -1} = \frac{x^2 - 1 + 1}{x^2 -1} = \frac{x^2-1}{x^2-1} + \frac{1}{x^2 -1} = 1 + \frac{1}{x^2 + 1}

 

[tony873004]

2 great answers. I knew it wasn't hard, but I'd have never come up with either of these on my own. Thanks!