GMAT Exam: A Simple Guide To Algebra - Rational Expressions

A rational expression is an algebraic fraction that has a polynomial for its numerator and a polynomial for its denominator.

For instance, is a rational expression. Because division by 0 is undefined, you must exclude values for the variable or variables that would make the denominator polynomial sum to 0.

For convenience, you can assume such values are excluded as you work through the problems in this section.

Reducing Algebraic Fractions to Lowest Terms
Fundamental Principle of Rational Expressions: If P, Q, and R are polynomials, then:
, provided neither Q nor R has a zero value.

This principle allows you to reduce algebraic fractions to lowest terms by dividing the numerator and denominator by the greatest common factor (GCF).

Before applying the fundamental principle of rational expressions, always make sure that the numerator and denominator contain only factored polynomials as shown in the following examples.

Multiplying Algebraic Fractions
To multiply algebraic fractions:

(1) factor all numerators and denominators completely, (2) divide numerators and denominators by their common factors (as in reducing), and (3) multiply the remaining numerator factors to get the numerator of the answer and multiply the remaining denominator factors to get the denominator of the answer.

You can leave your answer in factored form as long as it is completely reduced.

Here is an example.


Be careful! Divide out factors only. If a numerator or denominator does not factor, enclose it in parentheses. Forgetting the parentheses can lead to a mistake.

Dividing Algebraic Fractions
To divide algebraic fractions: Multiply the first algebraic fraction (the dividend) by the reciprocal of the second algebraic fraction (the divisor).

Here is an example.


Adding (or Subtracting) Algebraic Fractions, Like Denominators
To add (or subtract) algebraic fractions that have like denominators: Place the sum (or difference) over the common denominator.

Simplify and reduce to lowest terms, as needed. Here are examples.


When subtracting algebraic fractions, enclose the numerator of the second fraction in parentheses because you want to subtract the entire numerator, not just the first term.

Adding (or Subtracting) Algebraic Fractions, Unlike Denominators
To add (or subtract) algebraic fractions that have unlike denominators:

(1) factor each denominator completely; (2) find the least common denominator (LCD), which is the product of each prime factor the highest number of times it is a factor in any one denominator; (3) using the fundamental principle, write each algebraic fraction as an equivalent fraction having the common denominator as a denominator; and (4) add (or subtract) as for like denominators.

Here are examples.


A prime factor is one that cannot be factored further.

Simplifying Complex Fractions
A complex fraction is a fraction that has fractions in its numerator, denominator, or both.

To simplify a complex fraction: Multiply its numerator and denominator by the least common multiple (LCM) of all the fractions in its numerator and denominator: