Rational Expressions    Simplification of rational expressions, reducing to lowest terms
A rational expression is a fraction of which the numerator and the denominator are polynomials.
A rational expression is reduced to lowest terms if all common factors from the numerator and denominator are canceled.
To reduce a rational expression to lowest terms first factorize both the numerator and denominator as much as possible then cancel common factors, i.e., divide their numerator and denominator by common factors.
Examples:  Reduce the following rational expression to lowest terms.   factoring out the minus sign,
 a - b = - (b - a)
or
 - a + b = - (a - b)   Examples:  Reduce or simplify the following rational expression to lowest terms.      Addition and subtraction of rational expressions
To add or subtract two fractions with the same denominator, add or subtract the numerators and write the sum over the common denominator.
To add or subtract fractions with different denominators:
First find the least common denominator (lcd -the smallest number that can be divided by each denominator).
Write equivalent fractions using this denominator. Then add or subtract the fractions.
The process for rational expressions is identical.
Examples:  Perform the indicated operations and reduce the answer to lowest terms.    Multiplication and division of rational expressions
Use the formulas for the multiplication and the division of fractions. and Notice the two special cases of which should be aware of: and Examples:  Perform the indicated operations and reduce the answer to lowest terms.       proof  Note, since each second line should be subtracted, the sign of each term is reversed.
Solving complex rational expressions
A complex or compound rational expression has fractional expressions in its numerator, denominator or both.
To simplify complex fractions, change the complex fraction into a division problem, i.e., divide the numerator by the denominator.
Examples:  Simplify complex fractions.       Intermediate algebra contents 