Let the rational function is .

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

Observe the graph :

The -intercepts of the graph are  and (graph crosses the -axis; odd multiplicity).

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 even multiplicity in .

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

Since the polynomial has a horizontal asymptote then the degree of the numerator and denominator is same.

Thus, the numerator may contain a term.

So one possibility for the numerator is .

A possibility for the denominator is .

Therefore, one possibility for the rational function .

One possibility; .