Simplifying Rational Expressions

Brooks, Olivia, Diana, and Alexis

Notes

A rational expression is nothing more than a fraction in which the numerator and/or the denominator are polynomials. Here are some examples of rational expressions.


When we're given a rational expression, we can simplify it by doing the first two steps of polynomial division: we cancel any common factors, then factor the numerator and denominator and cancel anything else we can get away with.


1) Look for factors that are common to the numerator & denominator


2) 3x is a common factor the numerator & denominator. Note that it is clear that x ≠0


3) Cancel the common factor


4) If possible, look for other factors that are common to the numerator and denominator. In our example, we can use foil in reverse to factor an (x − 1) in the denominator and further cancel this binomial from both the numerator and the denominator.


5) After cancelling, you are left with 1/(x-1)


6) The final simplified rational expression is valid for all values of x except 0 and 1.
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Review Sheet

1. -36x^3/42x^2


2. -70n^2/28n


3. 45/10a-10


4. v-5/v^2-10v+25


5. x+6/x^2+5x-6

Review Sheet Key

1. -6x/7


2. -5n/2


3. 9/2(a-1)


4. 1/v-5


5. 1/x-1