Rational Expressions

Objective 1 Reduce a Rational Expression to Lowest Terms
If we form the quotient of two polynomials, the result is called a rational expression .
Some examples of rational expressions are

Expressions (a), and (b) are rational expressions in one variable , x , whereas (c) is a rational expression in two
variables, x and y.

When the numerator and denominator of a rational expression contain no common factors (except 1 and −1), we say
that the rational expression is reduced to lowest terms , or simplified.

The polynomial in the denominator of a rational expression cannot be equal to 0 because division by 0 is not
defined.
For example, for the expression, x cannot take on the value 0 . The domain of the variable x is {xl x ≠ 0}.

Example 1 Reducing Rational Expressions to Lowest Terms
Reduce each rational expression to lowest terms.

Objective 2 Multiplying and Dividing Rational Expressions

Example 2 Multiplying and dividing Rational Expressions
Perform the indicated operation and simplify the result . Leave your answer in factored form.

Objective 3 Adding and Subtracting Rational Expressions

and

are two rational expressions, then

Example 3 Adding and Subtracting Rational Expressions with Equal Denominators
Perform the indicated operation and simplify the result. Leave your answer in factored form.

Example 4 Adding Rational Expressions Whose Denominator Are Additive Inverses of Each Other
Perform the indicated operation and simplify the result. Leave your answer in factored form.

and

b≠0,d≠0 are two rational expressions, then

Example 5 Adding and Subtracting Rational Expressions with Unequal Denominators
Perform the indicated operation and simplify the result. Leave your answer in factored form.

Objective 4 Use the Least Common Multiple (LCM) Method

The least common multiple (LCM) of two or more polynomials is the polynomial of least degree and least positive
constant factor that has each of the given polynomials as a factor.

The least common denominator (LCD) of two or more rational expressions is the LCM of their denominators.

To find the LCM of two or more polynomials you first factor each of the polynomials completely. The LCM will be
the product of all the different factors , each factor occurring in the product the greatest number of times that it occurs
in the complete factorization of any one of the given polynomials.

Example 6 Finding the Least Common Multiple
Find the least common multiple (LCM) of the following pair of polynomials

and

Example 7 Using the Least Common Multiple to Add Rational Expressions
Perform the indicated operation and simplify the result. Leave your answer in factored form.

Example 8 Using the Least Common Multiple to Subtract Rational Expressions
Perform the indicated operation and simplify the result. Leave your answer in factored form.

Objective 5 Simplify Complex Rational Expressions
When sums and/or differences of rational expressions appear as the numerator and/or denominator of a quotient, the
quotient is called a mixed quotient (complex fraction).

For example,

and

are mixed quotients.

To simplify a mixed quotient means to write it as a rational expression reduced to lowest terms.

Example 9 Simplifying a Mixed Quotient (Using Method 2)
Simplify:

Example 10 Simplifying a Mixed Quotient (Using Method 1)
Simplify: