Rational Expressions


A Rational Expression is the quotient of two polynomials $ P$ and $ Q$ with $ Q\ne 0$.

Ex. $ \frac{x+1}{x^2+1}$.

Lowest Terms of a Rational Expression A rational expression is written in lowest terms when the greatest common factor of its numerator and denominator is 1.

Remark To write a rational expression in lowest terms we use the basic principle of fractions: $ \frac{ac}{bc}=\frac{a}{b}$, ($ b \ne 0$, $ c\ne 0$).

Ex. $ \frac{x^2-1}{(x+1)^2} = \frac{x-1}{x+1}$.

Operations of Rational Expressions For fractions $ \frac{a}{b}$ and $ \frac{c}{d}$ ($ b \ne 0$, $ d\ne 0$),

Multiplication and Division: $ \frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}$, and $ \frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}$

Addition and Subtraction: $ \frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$ and $ \frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}$.

The Least Common Denominator (LCD) of two rational expressions:

Ex. The LCD of $ \frac{1}{(x-2)(x+3)}$ and $ \frac{x}{x^2-4}$ is $ (x^2-4)(x+3)$.

A complex fraction is the quotient of two rational expressions.

Ex. $ \frac{\frac{1+x}{1-x}}{\frac{3+x}{4+x}}$



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Department of Mathematics
Last modified: 2005-08-30