Rational expression

A rational expression (in abbreviation as RE) is one of the forms of expressions in algebra. The rational expression definition is an expression which is in fractional form of two expressions. A rational-expression is generally denoted as p/q, q =! 0.. If ‘p’ and ‘q’ are numbers, then it is a fractional number. If ‘p’ is an expression with variable/s and ‘q’ is just a number, then the RE becomes as a simple expression with each coefficients of p is divided by ‘q’.

Rational-expressions have certain constraints which a normal expression may not. As the first one, we said the denominator expression cannot be zero. Thus, a rational expression can have domain and range restrictions. That is, a RE is not defined for the real zeroes of the denominator part. Thus, the domain has to be excluded for such values of the variable/s, and correspondingly the range is affected.

As a convention, the denominators of REs cannot be left with radicals, negative exponents or complex numbers. Putting back them in proper way is called as solving rational expressions. Let us discuss how to solve rational expressions in such cases.

In case of a RE with a radical term in the form (a + sqrt b) in the denominator, multiply both numerator and denominator by the conjugate of (a + sqrt b), which is (a - sqrt b). Now as per the ‘sum and difference product formula’, the denominator becomes as a2 – b, free from radical terms.

In case of complex numbers, the method is exactly same. This process is also called as ‘rationalizing the denominator’.
REs have more prominent place in rational functions. A rational function f(x) is normally expressed in the form f(x) = [g(x)/h(x)]. The domain, range, continuity, asymptotes are all dependent on the nature of g(x) and h(x). A rational function will have vertical asymptotes at the zeroes of h(x). Determining the horizontal asymptotes (HA) of a rational function is a bit lengthy but can me summarized as below. If the degree of g(x) is ‘m’ and that of h(x) is ‘n’, then,

If, (m – n) < 1, then the HA is y = 0, that is, the x-axis.
If, (m – n) = 0, then the HA is y = b, where ‘b’ is the ratio of leading coefficients.
If, (m – n) = 1, then there is no HA but the slant asymptote, is y = mx + b, where ‘mx + b’ is the quotient part of the long division of g(x)/h(x).
If,(m – n) > 1, neither HA or slant asymptote.