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 (sometimes called an algebraic fraction) 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.