Let the rational function is .

The numerator of a rational function in lowest terms determines the -intercepts of its graph.

observe the graph :

The -intercept of the graph is (even multiplicity; graph touches the -axis).

So one possibility for the numerator is .

The denominator of a rational function in lowest terms determines the vertical asymptotes of its graph.

observe the graph :

The vertical asymptotes of the graph are and .

Since approaches to the left of and approaches to the right of , is a factor of odd multiplicity in .

Since approaches to the left of and approaches to the right of , is a factor of odd multiplicity in .

A possibility for the denominator is .

Therefore, one possibility for the rational function .

One possibility; .