Many algebraic fractions are rational expressions, which are quotients of polynomialswith nonzero denominators. Examples include

Properties for working with rational expressions are summarized next.

PROPERTIES OF RATIONAL EXPRESSIONS

For all mathematical expressions P , Q , R , and S , with Q and S 0.

Fundamental property

Addition

Subtraction

Multiplication

Division

When using the fundamental property to write a rational expression inlowest terms, we may need to use the fact that

For example,

Reducing Rational Expressions

EXAMPLE

Write each rational expression in lowest terms, that is, reduce the expression as much as possible.

Factor both the numerator and denominator in order to identify any commonfactors, which have a quotient of 1. The answer could also be written as 2x + 4

The answer cannot be further reduced.

CAUTION

One of the most common errors in algebra involves incorrect useof the fundamental property of rational expressions. Only common factors may be divided or “canceled.” It is essential to factor rational expressions beforewriting them in lowest terms. In Example 1(b), for instance, it is not correctto “cancel” k (or cancel k, or divide 12 by -3) because the additions and subtraction must be performed first. Here they cannot be performed, so it is notpossible to divide. After factoring, however, the fundamental property can beused to write the expression in lowest terms.