Simplifying Rational Expressions
As you know, a rational number is one that can be expressed as a fraction, that is,
p/q,
where p and q are integers (and q ≠ 0).
Similarly, a rational expression is one that can be expressed as a quotient of polynomials, i.e. p/q where p and q are polynomials (and q ≠ 0).
Example:
is a rational expression, since both the numerator and the denominator are polynomials. ("3" counts as a polynomial... it's just a very simple one, with only one term.)
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.
Example:
Simplify.
First, factor out a constant from both numerator and denominator. Write the 9 as 3 · 3.
Next, factor the quadratic in the denominator. (Look for two numbers with a product of –6 and a sum of –1.)
Finally, cancel common factors.
IMPORTANT NOTE: EXCLUDED VALUES
When we factored out x + 2 in the above expression, we made an important change. The new expression
is defined for x = –2; it equals –1/5. But the original expression we were trying to simplify,
is undefined for x = –2, 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.
