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 q0 ).

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 q0 ).

Example 1:

3 x 3 +5 x 2 y7 y 3

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.)

5x+3 6x+ x x y

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 2:


3x+6 9 x 2 9x54

First, factor out a constant from both numerator and denominator. Write the 9 as 33 .

3x+6 9 x 2 9x54 = 3x+6 33( x 2 x6)

Next, factor the quadratic in the denominator. (Look for two numbers with a product of 6 and a sum of 1 .)

= 3(x+2) 33(x+2)(x3)

Finally, cancel common factors.

= 1 3(x3)


When we factored out x+2 in the above expression, we made an important change. The new expression

1 3(x3)

is defined for x=2 ; it equals 1 5 . But the original expression we were trying to simplify,

3x+6 9 x 2 9x54

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.