Rational Functions

rational functions, factoring, asymptotes, rational numbers, factorable

In an analogous way to rational numbers, which can be expressed as the quotient of two integers, rational functions can be expressed as quotients of polynomial functions. And by quotient, we mean an integer divided by another integer, or in the case of rational functions a polynomial function divided by another polynomial function.

Thus, suppose that we have two polynomial functions, P(x) and Q(x).  Then, we can create a rational function H(x):

H(x)=P(x)/Q(x).

Suppose we have P(x)=x²+4x+4 and suppose we have Q(x)=x²+5x+6. So we can create a rational function as follows:

H(x)=(P(x))/(Q(x)), or H(x)= (x²+4x+4 )/(x²+5x+6)

(1) Because both Q and P are factorable, we can factor H(x). Also, if we find common factors of P and Q we can reduce the rational function.

(2) Factoring rational functions also allows us to easily find the zeros of Q(x), the denominator in the rational function.  From this, we can figure out where the rational function is undefined (which amounts to finding what is not in the domain of the rational function).  This is also the very means by which we can figure out the asymptotes of the rational function, which we will go into in more detail in other topics.  Specifically, finding asymptotes.

rational functions, factoring, asymptotes, rational numbers, factorable