Simplifying Rational Expressions
where and are integers (and ).
Similarly, a rational expression (sometimes called an algebraic fraction ) is one that can be expressed as a quotient of polynomials , i.e. where and are polynomials (and ).
is not a rational expression. The denominator is not a polynomial.
A rational expression can be simplified if the numerator and denominator contain a common factor .
First, factor out a constant from both numerator and denominator. Write the as .
Next, factor the quadratic in the denominator. (Look for two numbers with a product of and a sum of .)
Finally, cancel common factors.
IMPORTANT NOTE: EXCLUDED VALUES
When we factored out in the above expression, we made an important change. The new expression
is defined for ; it equals . But the original expression we were trying to simplify,
is undefined for , because the denominator equals zero (and division by zero is a no-no).
So, our simplification is not really true for all points. When you simplify rational expressions, you should make note of these excluded values .