Finding when a rational function is positive

To find when a rational function is positive what you should factor so that both the numerator and the denominator are products of simple pieces. Then look at the signs of the pieces and use the fact that a product with an even number of negative factors will be positive and with an odd number of negatives will be negative. It is easy to see where linear factors change sign. Irreducible quadratics don't change sign, so they will be positive if the leading coefficient is positive.

For example to see where

$$f(x)=\frac{(x+1)^3x(x^2+1)}{(x-1)^2(x^2-4)}>0$$

we would first complete the factorization to get

$$f(x)=\frac{(x+1)^3x(x^2+1)}{(x-1)^2(x+2)(x-2)}>0$$

and then notice that the places where the sign could change are where $x$ equals -2, -1, 0 , 1, and 2. We make a table with a line for each factor.

$(x+1)^3$

-

-

+

+

+

+

$x$

-

-

-

+

+

+

$(x^2+1)$

+

+

+

+

+

+

$(x+2)$

-

+

+

+

+

+

$(x-2)$

-

-

-

-

-

+

$(x-1)^2$

+

+

+

+

+

+

$f(x)$

+

-

+

-

-

+

$x$

-2

-1

0

1

2

We conclude that $f(x)>0$ for $x\in(-\infty,-2)\cup(-1,0)\cup(2,\infty)$ and that $f(x)<0$ for $x\in (-2,-1)\cup(0,1)\cup(1,2)$.