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Simplifying Rational Expressions
A rational function is a function that looks like a fraction and has a variable in the denominator. The following are examples of rational functions:

Note that a function such as is not considered a rational function. Even though it is in the form of a fraction, the denominator does not contain a variable.

Simplifying rational expressions usually involves factoring. Once the expressions have been factored, you can usually divide out common terms and write the expression in its most simplified form.

All of our examples will have terms that can be divided out. But be careful, this may not always be the case. Some rational expressions will not simplify. Also be aware that sometimes you may need to factor out a "-1" in some situations that involve terms in the form: (a-b) and (b-a), as in Example #4 below.

#1: Simplify .
  • Begin by completely factoring both the numerator and the denominator.
  • Divide out any common terms that are in both the numerator and the denominator.
In our case the term: (x+3)
What is your answer?
#2: Simplify .
  • Factor the numerator and denominator.
  • Divide out common terms.
In our case both the factors 3 and one of the x's
What is your answer?
#3: Simplify .
  • Factor the numerator and denominator completely.
  • Divide out common terms.
What is your answer?
#4: Simplify  .
  • Factor the numerator and denominator.
  • The terms and cannot divide out at this point even though they are very similar. If we factor out a –1 from the denominator, we can continue the simplification.
What is your answer?

S Taylor

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