Easy algebra manipulation, example $x^3/(1+x^2)$

Question

I guess this is really basic but have trouble following some algebraic manipulations on fractions.

For example with two cases $$\frac{x^3}{1+x^2}$$

is supposed to be: $$x -\frac{x}{1+x^2}$$

and this identity $$\frac{x^2}{1+x^2}$$

is supposed to the same as this: $$1 -\frac{1}{1+x^2}$$

Which steps do you come to these conclusions, my guess is that you add or multiply something to the nominator and denominator but what and what type steps and what type of thinking is behind?

Thanks

Answer 1

Concerning the first example, note that\begin{align}\frac{x^3}{1+x^2}&=\frac{x+x^3-x}{1+x^2}\\&=\frac{x(1+x^2)}{1+x^2}-\frac x{1+x^2}\\&=x-\frac x{1+x^2}.\end{align}Can you deal with the other example now?

Answer 2

Write your term in the form $$\frac{x+x^3-x}{1+x^2}=\frac{x(1+x^2)-x}{1+x^2}=x-\frac{x}{1+x^2}$$