Rational expressions are usually simplified so that the lowest common denominator appears. Reduction of rational expressions to the simplest form is performed by eliminating common factors that appear in both the numerator and denominator. We saw some examples of this above. The set of common factors is the GCD for the numerator and denominator. This is an application of the fundamental principle of fractions in which the numerator and denominator of the rational expression are both divided by the GCD. Example:
A rational expression can sometimes be reduced to a polynomial with negative exponents. In the example below, the fundamental principle of fractions is applied by multiplying the numerator and denominator by the reciprocal of the denominator:x2 + 3x + 2 (x + 2)(x + 1) x + 2 ————————————— = ——————————————— = ——————— x2 - 1 (x + 1)(x - 1) x - 1
xy - 1 ——————— = xy-1 - y-2 y2
Complex fractions are expressions containing fractions in the numerator and/or denominator. They can be reduced to simple fractions by applying the fundamental principle of fractions and the operations shown above. Example:
x y x2 - y2 ——— - ——— ———————— y x xy x2 - y2 ———————————— = ——————————— = ———————— x y x2 + y2 x2 + y2 ——— + ——— ———————— y x xy